Setting yourself up as a medical expert would not necessarily be your best choice.
You will be in opposition to the dominant medical experts of the day. If you ever fail to produce the desired outcome, your unconventional practice will certainly be used against you. Some kings were known to behead failed doctors, etc. even close friends and family. If you are successful, your enemies may accuse you of sorcery.
Improvements such as soap and sanitation would not necessarily be popular as the value would not be a priori considered valuable. So getting a king to implement this and recognize this as successful may be very difficult. A crude optical microscope and basic germ theory may again just be witchcraft.
As a practicing physician your best bet may be to use the technology of the time, i.e., bloodlettings, etc. but supplement them with curatives of your own design that you don't bother to mention. Losing the occasional patient will still be be par for the course, but if you don't upset others by your unconventional medical practice, your better than average success rate will serve you and your patients well. Maybe by the time you get appointed chief court physician you can try some of your wacky ideas of sanitation, etc.
The scientific method was not exactly understood at this time. But advantages in science and engineering were generally valued by the educated, which certainly included the royalty and the clergy. You would not share the scientific understanding needed to understand new technologies, but your patron would certainly expect you to share the methods of production.
Just going into production of gunpowder would not make your patron generally satisfied. You would also need to produce cannons that could use the gunpowder (probably bronze unless you wanted to invent steel-making too). And even if you produced cannons and gunpowder, your patron would want to be able to produce them himself so as to not be 100% dependent upon you.
If you don't want to reveal tech, you pretty much cannot reveal it in small doses. Smart observers will pick up on your unconventional ideas that force you into areas you would rather not pursue.
I am a big fan of alternate histories. Lord Conrad Stargard of the Cross-Time Engineers series is essentially in the position of your character without the medical training. He ingratiates himself to the king by full scale engineering - so that he can repel the Mongol invasion of 1240 he is expecting in 10 years. Conrad has little choice - either prepare full scale to repel the Mongols or flee to another area not to be invaded. Since he is a Polish patriot he makes his stand. One thing I like about the series is that it takes time for Conrad to be accepted by the various layers of royalty and for them to put their trust in this outsider. Series had serious weaknesses (later books were pretty bad) and his sexist / racist views are pretty grating. His luck and assistance from the future can also be annoying, but the way be works with the royals is just what an engineer would do.
One way in which Conrad was extra lucky is that he just happened to be carrying a large variety of modern, non-hybrid seeds when he got sent back in time (part of trying to impress a girl). These seeds had a huge advantage compared to historical yields where you often had to plant 20% of your crop for next year's harvest. If you are actually planning your trip, such seeds would also be to your advantage, assuming you can dodge being labeled as a devil-worshiper.
Sorry, have to share one other issue with medicine. If your medicine is in the form of drugs, you may have a problem - route of administration. Injections are clearly not going to be available. Oral administration may be complicated by the corrosive stomach acid. Many medicines require special coating, etc. to get past the stomach, coatings that will not be available. Fortunately, enemas are often a useful route of administration, and as luck would have it, common practice in medieval medicine though as you might expect not as sanitary as you would like.
There are several other technologies that other people have mentioned, though you wanted to avoid sharing technology. The invention of distilled alcohol might be one you would be willing to share. It was already invented by the Muslims, but had not spread to Europe by 1000 AD. So, you could invent it in Europe and realizing the commercial value set up a distillery. Start with fortified wines, graduate to ports and sherries, and eventually hard liquor and you have customers coming to you for years looking for something new to drink. Royalty would come looking for you as a purveyor of fine spirits.
There may be other technologies that you would be willing to share. Not sure why you are not willing. But eyeglasses. mechanical clocks, the spinning wheel, the stern mounted rudder, paper, printing press, the wheelbarrow and others were all waiting to be discovered or introduced to Europe in 1000 AD. Perhaps some of these are things you would be willing to allow outside of your control.
If you really want to control kings, there is the tried and true method. Put them in your debt literally - i.e., become a banker known for generous loans to kings. You will want to become rich very quickly, but given your inventive nature this should not be too hard.
You have a pretty fundamental problem, trying to do this all in a few weeks or months. Getting on the good side with local royalty, etc. is likely impossible in a short time-frame. Your proposed saving the life of the King's daughter is probably unrealistic, the king will not allow you access to his daughter. It is unlikely you would even be allowed to talk to the King. His guard, counselors, etc. are designed as a filter to restrict access. To get access quickly you have to ingratiate yourself to someone else that has access to the king, but they them-self are not particularly hard to access. A member of the guard or household staff might have access (possible indirectly) to the king, and a miraculous cure of one of their family might grant you the inside track.
Such an approach does not give you the inside track to a number of royalty and clergy though. If you manage to get access to a single king in this way, he may treat you are a proprietary resource and not allow you access to anything outside his royal court. To guarantee access to a number of members of royalty you need broader influence that would make it harder for the king to control you. Of course, this argues directly against gaining access in such a short time-frame.
This popped into my head, and it would be of great value to kings and such-like. You just "invent" some pseudo-modern cryptography. You need to choose something much better than they are used to, but that can still be encrypted and decrypted by hand assuming you know the key, but without a computer you will not be able to decrypt without the key. You want something stable in the sense that errors do not propagate so that a mistake does not garble the rest of the message since encryption by hand is error prone. Cryptography was not even a subject of significant study at this time, so some kings would have to have the value of this explained to them, though they are likely familiar with the Caesar Cypher. Kings will understand the value very well once it is explained to them.
