You have two issues to deal with at the same time: to improve your health and to resolve the issues with your residency. On your health, you need to talk to your medical practitioner to ensure that you understand your prognosis, the possible treatments and their outcomes, and the timing. That will give you the information you need in order to deal successfully with the issues with your residency.
On your residency, you can either try to carry on, ask for a LOA, or resign.
If you try to carry on, you need to be certain that you will be able to remediate your performance to a standard and timescale acceptable to your program. If you want to go down this route, first take a look at your contract with the program to see where you stand in terms of warnings, remediation and probation. If possible, find a resident advocate within your residency system and talk to them about what to do, and have them accompany you to meetings. (If there is no-one available within the formal system, find a sympathetic attending or senior resident who is knowledgeable about the program and would help.) Then discuss where you are with the person in the program who has the power to sort out a way forward for you which will bring you back into good standing with the program.
If you want to carry on with your residency, it would probably be best if you gave your program an explanation for your below-par performance - it will help them to understand why things are not as they should be, and hopefully then to work with you to put them right. In your case, your explanation is that you have an illness which has lasted for a certain period of time at least and which is, at present, still ongoing and for which you are receiving treatment. How much more you say about what that illness is will be a matter of judgment for you.
If you want to take a leave of absence, again the first thing to do is to check what your residency contract says about this, and then to talk about your situation with someone you trust who is knowledgeable about the program and will be on your side. If you suggest a leave of absence to your program, both you and the program will need to be clear about your route for returning and completing your residency. Make sure that you get the terms on which you are taking the leave of absence in writing, including a written commitment to the terms on which you will return and complete your residency.
Resignation is pretty much your last resort. If you do intend to resign, you should negotiate the terms of your resignation with your program before you actually resign. Those terms can include things like the amount of credit you will be given for time already spent in residency, the references the program will give you, and the amount of help they will give you in finding another residency, either immediately or after a period of recuperation. Your leverage for those negotiations will not be great, but will include the program's desire to do the right thing by you and to maintain its reputation.
Good luck.
