I'd avoid sending her a letter in which you ask her out if I were you. If you want to leave your contact information at the end, go for it, but don't try to ask her out or declare your interest in the card. That would be bad form.
Medical professionals appreciate food gifts. A gift basket might be better appreciated than a box of chocolates, though medical professionals are getting less chocolate these days. A gift basket can be divided up among staff and brought home instead of sitting in the break room forever. And I personally hate the female "Oh, I shouldn't have another, or even one" "What?! No, you can have as many as you like. I'm the one who should stay away" dance. Of course, you won't have to witness it.
Letters are so rare, they're almost creepy these days. [Yeah, yeah, yeah. I know. This is AoM. Bring back the romance. Sorry. Some ships have sailed.] A typed letter to all the staff thanking them for their care and specifically calling out this particular nurse will be VERY appreciated however. Best to address it to the boss and ask that it be posted in the break room. (This way, there's a professional, not just emotional, boost.)
As for the particular nurse and romance, I agree with the others that this has probably happened before, but that probably means she knows how to deal with it. She is free to date you in terms of professional ethics, but she may have other personal standards. Or she may not be interested for any of the usual reasons. I'd say wait awhile after sending the gift/letter, then call her (at work, but you can't reach her otherwise) and ask her out. If you just add your number to a card, that's too subtle.
I hope the transplant works out for you, and I'm impressed to see that you are in high enough spirits to think thoughts of romance.
I understand how you might have gotten close to your healthcare providers because of the circumstances. But as Rick Shelton pointed out, the situation was not symmetric.
I agree with your other respondents that asking her out would not be appropriate. A modest, non-personal gift, perhaps a box of goodies to eat she could share with her colleagues, and perhaps a letter of thanks and appreciation would be right. That would be tasteful and considerate. That means WITHOUT anything that fishes for "whether she is interested in me in that way or not". Yes, you could DISCRETELY include contact information if she wanted to contact you, in which case you would be letting her make the choice without being pushy. But appreciate the patient-caregiver relationship for what it was, without unrealistic expectations.
If you pressured her, and she succumbed to pressure but did not have any personal feelings for you, that could lead to heartache for both of you.
Good health to you and great success in matters of love.