I would think about your friend as you write, rather than your own pain. Your friend might be in pain too. Sometimes things are too hard for us, I had a hard time visiting my father in the hospital and my stepmother in the assisted living center, etc. Maybe your friend had a hard time because of something in her past with your not doing well and going to hospital; maybe she fears for herself. I would concentrate in the letter on her and trying to make her "comfortable" with you and the friendship again (if you want to keep up the friendship). I'd be a little lighthearted, suggest meeting somewhere neutral as if you're just meeting again.

What your friend did (do you know for sure she was home and heard you knock (did you knock? I got into a sad time with a friend for a year because her doorbell was broken) when you went to visit) was painful for you but if you concentrate on that you don't "understand" rather than how you are hurt, it might help both you and her get together. She could be feeling very guilty that she wasn't there to help you when you were in the hospital. Telling her how much you were/are hurt will only make her feel more guilt and the only thing she can do about that is stay away from you and thinking about you. What happened in the past is in the past and needs to be moved beyond or closed. I would try to be reconcilitory and move the relationship forward/back to stable ground in the first letter and then if she doesn't show up where you suggest meeting or something, then I'd maybe write another to close it. I try not to end relationships myself with people I really like in case some time a lot further down the road we start up again. My friend and I her were apart a year are doing fine now and talk to/see one another occasionally again. I have another friend who dropped me (and everyone else) but I don't do anything in case she wants to pop up again. I miss her but there's always a bit of hope we can get together again. People come and go like that in our lives; no one stays forever.