This could be fun, so I will try to write a gentle answer.
The overall picture is simple...
Political power will eventually tend to shift to class or classes which have economic power. That is because in any society it is the economic base which supports and generates the beautiful and intricate superstructure.
Remember the golden rule: he who has the gold makes the rules. (Not necessarily literal gold: the "gold" could be silver, as in Ancient Greece; or it could be land, as in the European Middle Ages; or it could be capital in general, as in the modern world. But you get the picture.)
The overall picture is just that, an overall picture. In the real history of real societies the shift of political power towards the class or classes with economic power was never quick; it usually did happen eventually, but there is always a considerable amount of time between the moment when, let's say, merchants and tradesmen become the economic engine powering the society and the moment when the Peers and Barons cease to
be the dominant political factor.
... But details are essential
The essential point is that there is no such thing as "the" the most important factors that determine the balance of power between kings and nobles and merchants at any given point in time. Yes, in the end political power will tend to shift to the class or classes wielding economic power; but this shift may take decades or even centuries, and it may or may not necessitate a violent revolution.
That is because the balance of power between kings and nobles and merchants is, to use the Marxist phrase, historically determined, that is, it fundamentally depends on the specific situation:
It depends on the history of the society in question.
For example, the Emperor of the German Empire still had enough power at the beginning of the 20th century to engage his empire into a suicidal war againt the entire world.
(Spoiler alert: they lost.)
The point being that in 1914 Germany was a very highly developed nation-state, a world leader in science and technology, one of the most important exporters of expensive industrial products, an example of a stable society, with fully acknowledged rule of law etc. But their emperor was still enormously powerful, and only because of the history of how the German Empire had come into existence.
For another example, compare England and France around 1700. The two countries had just about the same level of scientific, technological, and economic levels of development. Both were reasonably great powers (as great as powers could be at the end of the 17th century). Both were stable societies. Both were convinced that they were the pinnacle of human civilization. And yet, the dual sovereigns William and Mary of England had very much more limited powers than Louis XIV, the absolute *"l'état c'est moi" king of France.
What do we learn from those examples? We learn that whatever list of factors we can come up with it won't be able to address any specific situation without taking into consideration the history of how the society arrived in that situation.
For an example in our present time, compare and contrast the power of the President of the USA with the power of the Chancellor of Germany; both countries have a federal structure, both are highly developed, both are great economic powers. And yet the POTUS is an almost absolute king (term-limited, mercifully), while the Chancellor of Germany must constantly manoeuver to keep the Bundestag happy.
It depends on idiosyncratic accidents.
Sometimes, actually quite often, the balance of power between the sovereign and the nobles (and even other political forces) depends on unforeseeable accidents. Is the king smart, determined, with a strong character, or is he a dullard who likes to live the good life and doesn't care about actually ruling? Does the great king die without a heir? Do the nobles have a strong leader? Is there religious strife in the land? Was there a seven years drought bringing the peasants on the brink of revolt?
For an great example of how a country can abruptly transition from an absolute monarchy to the absolute domination of the great nobles, consider what happened in Hungary when king Matthias Corvinus died and the nobles, in a moment of epic selfishness, elected the hapless Vladislaus II of Bohemia as king. (They did get rid of royal absolutism and got to enjoy 36 years of freedom from royal rule; but in 1526 the Turks taught them a signal lesson on the foolishness of their actions; after the battle of Mohacs, the entire issue of the balance of power between king and nobles became irrelevant in Hungary.)
It depends on who has the economic power.
Is there a numerous and rich mercantile class, generating most of the tax revenues? If so, the king may well prefer to rely on the merchants insted of on the nobles. Is the Crown really really rich? If so, a determined king may well choose to impose authoritarian absolutism. Is the economy dominated by agriculture? If so, the great landowners will almost certainly rule, either directly or indirectly.
Plus that pesky historical determination raises its head again. Does the country have a tradition of not having a large standing army? Then the king will almost certainly lose power in favor of either the merchants or the nobles, depending on who is richer and can muster an actual efficient fighting force; see for an illuminating example the history of Cromwell and the Commonwealth of England.
It depends on what the dastardly hereditary enemies do.
Countries are never alone, they always exist in a complex web of international relationships. And sometimes it happens that the devious plans of the hereditary enemy succeed beyond their wildest hopes, drastically altering the balance of power between the king and the nobles great and small.
For example, by a perverse course of historical accidents, by the middle of the 17th century the Polish-Lithuanian Commonwealth had come to adopt a rule of parliamentary procedure which required unamimous consent: the liberum veto. It didn't take long for Russia to realize that it was enough to bribe one Polish nobleman, preferably the poorest one as requiring the smallest bribe, in order to paralyze any kind of decision making in Poland. The problem of the balance of power between king and nobles went away because there was no longer any power to balance. By 1795 Poland itself was no longer, its territory incorporated into Austria, Prussia and Russia.
The conclusion is that it's complicated. Anybody who tells you that you can have a nice little list of factors which will tell you how power is balanced between king, nobles and merchants at a specific point in time is either an idealist to whom messy details don't matter, or else wants to sell you a book.
Instead of looking for a list, look at the real post-medieval history for clues and examples, and model your society on one or more of the real-life kingdoms. (And republics! There were a handful of republics, notably Venice, which in those days was a reasonably great power).
And really really try to read Montesquieu's Spirit of the Laws. (The link goes to Archive.org; the second volume is also available there.)