About Feudalism (Section Edited)
In Feudalism, we start at the top with the ruler, who is essentially a landowner. He or she then doles out parcels of land to lesser lords and/or ladies, who are then responsible for the occupants thereof. Depending on the size, you could have many layers, from King, to Lord, to Dukes, to Barons, Counts, Viscounts, even Marquises, and so on and so forth.
Initially, the Feudal Lord ensured that farmers would be able to farm in peace by keeping a standing force and sending them out to kill bandits, rival forces looking to claim territory, and even packs of wolves or bears that threatened crops and livestock. The knights who became lords were the only ones capable of fighting off these threats, and so therefore became landowners with "tenant farmers".
In exchange, the farmers handed over their crops or livestock, as a "rent" or "tithe" or something of that nature. They paid for the privilege of using the land with goods.
The only laws were those made by the landowners, and only the laws that could be enforced physically were those that were considered true laws. Over time, the knight became "king" and was able to then dole out land to his trusted family and friends. However, the whole plot of land (the "fiefdom" or "kingdom") was still his and he held dominion over it.
Since this is your world, you don't have to follow the strict timelines of the feudal system in our world, but I would like to point out that by the time cannons rolled around we had largely abandoned the feudal system in favor of the monarchy, which is entirely different.
EDIT: See the comments below for a more realistic description of feudalism. In the early stages, there was no cash-based economy, therefore Kings were forced to "pay" their knights in land, which the knights then lived off of. This causes a problem as the more land the king gives away, the poorer he becomes, and he has no way of collecting anything from the land he gave away. The knights can then use their land and power to usurp the king, causing instability and general problems which went on throughout the feudal period.
As you had initially requested something "genuine" rather than "factually accurate", I outlined a cash-based feudal system below, which isn't entirely historically accurate. Further research into the issues of the feudal system might benefit you if you want it to be more historically accurate, but hopefully some aspects of my answer will still apply.
Responsibilities
Your lady's primary duty is to make sure that her people, whether they be villagers on the edge of her territory or villages closer to the center, are safe. In addition to physical threats like those listed in the last section, this may also entail watching for plague-bearers, monitoring trade caravans that come through her land, keeping an eye on the weather, and watching for stages of drought or blight.
Vassals are given land to monitor because their Liege or King is too busy/rich/occupied to watch over every parcel of land. I liken it to management in the corporate environment. Your King and his advisors are the Chairman and his Board of Directors. They demand that profits continue to rise, and that their positive reputation is maintained. In order to do that, each "department" or parcel of land must produce money, through taxes, tithes, or something similar. This job falls upon upper management, who must ensure each staff member (the peasants) is performing efficiently, paying rent, and farming as much as possible so they can be taxed. Her responsibility to her king or liege, therefore, is to pay him the profits of the land, whether that be in taxes or tithe, or "donations", and to report to him that everything is going well.
Your lady can actually have as much or as little control as she pleases, but her goal is to maintain the population. Peasants have no ties to the land that they farm, so if a plague, blight, or poverty prevent them from working or surviving, they will begin to die off or will emigrate elsewhere. If she exerts too much control, demands too much in tithe, or on the other end of the spectrum, if she refuses to guard against attacks, or refuses to maintain certain things, she will find that her lands will be deserted, she will have no one to tax, and therefore no income.
Ideally, she wants to establish a safe and stable economy. Responding to immediate physical threats quickly will show the peasantry that she cares for their welfare. Day to day she should open herself or her stewards to inquiries from the people. Do they need to dig an additional well in order to get more irrigation to crops? Do they need paved roads in order to transport their produce to the village to sell? These concerns must be weighed based on their costs.
In this time there were no police, so she and her guards act as the judicial system, doling out justice, punishing criminals, and judging civil and criminal cases. Does she choose to judge each case herself, or are they too numerous, so she employs magistrates or guards to judge? She can use this as secondary income in the form of fines and reimbursement for these services, but may end up imprisoning some or forcing them to pay their fines with hard labor (this is a good way to get those paved roads without paying too much extra).
Any problems that arise that cannot be solved by her, must be passed along to her "supervisor" to be handled. If her troops are not sufficient, she must call in the King's Army. If a blight destroyed her crops, she must request food. If a plague arrives, she must request medicine. These things are provided to her by her king because she pays her tithe and taxes, just as she provides these things to her people because they pay her tithe and taxes. If she has a poor fief or hasn't been paying enough/regularly, the king may ignore her requests and leave her to suffer.
I hate to say it, but your vassal is essentially a Bookkeeper. She will spend most if not all of her days accounting in log books, or reporting to stewards who will record everything in log books. Evidence shows that modern accounting was invented in the Middle Ages, and many of these log books give us the best view into how the feudal era and the Middle Ages ran. Day to day she must track everything, or she must employ people who track everything.
How much produce is coming from the farms? Is it enough to keep some for winter? Can we sell some of it to neighboring fiefs for extra cash? Is the king making any special requests for produce (ie. is he having a birthday ball and needs 80 fresh geese?) How much income are the peasants making? Therefore, how much tax or tithe can she impose upon them? Do you demand a set sum of money each month/quarter/year or do you instead calculate a percentage of their funds? What if they can't pay? What if you usually demand a set sum but you notice a few peasants are making huge stacks of profit? Can you impose extra tax on them without them getting pissed off?
Out of this initial tax and tithe you take out what cost she needs to maintain roads, wells, bridges, and whatever other infrastructure she needs. Then she will take out how much she needs to maintain her household. She will have troops that need to be paid, fed, and armored, servants that need to be paid, fed, and clothed, horses and whatever other animals she might want that need to be cared for, plus all those magistrates, stewards, accountants and whoever else she employs to keep the work from overwhelming her.
Then her obligation must be made to her king or her liege. Whoever is next highest up in the food chain must be paid his/her taxes or tithe. This may be a percentage of profits or it may be x amount minimum, depending on the king's wishes.
Privileges
Only once her land, her people, her household, and her liege were "taken care of" in terms of taxes or tithe would she then get to take her "privilege". The remaining money would go to her own coffers to clothe, feed, or throw away on solid gold bathtubs as she sees fit.
In addition to the money, she would be considered very trusted by the king. The privileges this could garner her are too numerous to list here, but she would have titles, meaning she would likely be able to marry nobility if she chose. She would be permitted into the palace, perhaps even become one of the King's advisors herself.
Obligations to Liege
Now, this is the ideal state of things, wherein the vassal is responsible and caring and doing the "right thing" by meeting her obligations. However, there is no law saying that she has to do anything except meet the obligations to her liege.
He is the one with more power than her, and he can make her suffer. Like I said above, only the laws that can be enforced are considered true laws.
So long as she can provide him with his tax or tithe, she can spend the rest of the money as she sees fit, like on the aforementioned gold bathtubs, and she can tax her people into oblivion so long as they don't die or abandon their lands. Again, it's all about how nice you want her to be. The accounts of corrupt lords are numerous, but it is important to note that she might be inviting anger from the king, a peasant revolt, or worse by these actions.
In addition to the tax and tithe, the king may request extra produce or food (like for the aforementioned birthday gala), or he may request she send food to fiefs who are suffering from blight or drought. He may also summon her troops away from her lands if a national crisis occurs that requires more troops than he has in his standing army, and he can recruit her peasants as makeshift warriors if it is a truly huge war and he is desperate for soldiers.
There is a ton more to be said, I would recommend looking at the links I shared, as well as looking into feudalism in general. There are a wealth of essays online, and like I mentioned above, we have a great view into feudalism, taxation, and other items like this because of the accounting work that went into this time period.
Take a look here for some common misconceptions about feudalism as well.
