I would not go into any detail with the professors. I would just say you had health issues the last couple weeks such that you were in the hospital and could you have X extra days to make up the work or hand in an assignment or whatever you want to ask for. Be specific about what you want to ask for and have a backup plan if they don't go with what you want primarily, something like you'd like an incomplete please (and finish/fix it over the summer).

I'm assuming you have been doing okay in your classes up until the last few weeks? If you have been spotty all semester, I would just give up that class and retry it later.