OP, I think people are being too hard on you. I would have let the pay advance request go too - I wouldn't have given it, but I don't think I would have let her go then, if my gut was telling me otherwise. (Of course, my gut thinks good of everyone, but that's another issue . . .
However, I agree with others that you do need to let her go now. I would just be worried about her health and ability to care for a newborn. What if something happened to her while she was working? An older child would be able to use the phone to call for help and not kill himself while he waited for it, but an infant is completely at the mercy of his caregiver. What if your nanny has another mini-stroke and passes out at 10 am one day? You might not know anything was wrong and your baby was alone until you got home that evening! Frankly, there is nothing that she or her doctor could say that would convince me to take this risk.
That said, I wouldn't add to the nanny's stress by letting her go while she's in the hospital. I agree with previous PPs who suggested asking her daughter for the parking pass and keys, to pass on to your "temp nanny." Then, when your nanny is out of the hospital, and tells you she's ready to start, gently tell her that you're just not comfortable given her fragile health and the demands of a newborn.
If you get no response from the daughter, wait till nanny's out of the hospital, ask to meet her at a Starbucks, buy her a coffee, let her go, and get your stuff back.
Good luck!