I'm not going to say, "No, don't!!!", but I will be a bit of a rain cloud:
Right now, you may be looking good to her because she's in the hospital with a lot of time on her hands and not a whole lot to do with it, and you just happen to be there frequently interacting with her. Plus as you said, she's far away from home and lonely. Perhaps there's a spark that she really feels...or perhaps she's just amusing herself with some flirtation to pass the time.
And she may be looking good to you because she's showing (or you think she's showing) this interest, and because you get to spend several hours a day with her, getting pretty much her exclusive attention. Perhaps there's a real spark you're feeling too...or perhaps it's just a feeling of excitement because she's livening up what would be an otherwise dull work week.
Once she's discharged and you're not around each other all shift, she--and you--might feel differently about things. So unless she asks you out first, definitely wait a week or so after she is discharged to do anything. Then, if she's still on your mind, do the "bump into her" bit.
If a relationship was meant to be, it will happen regardless of whether you ask her out now or at the Y in a week/month/whenever. If it wasn't meant to be, you'll either have forgotten about bumping into her, or will probably get the message in the first bump or two.
Sorry to be that rain cloud, but better you consider the possibility now before you go any further and risk embarassment on either/both parties