Leaving the hospital with unpaid bills.

Hello. I have a sister that is detained in St. Lukes Hospital right at this moment. I say detained because she is now capable of leaving but cannot because of an unpaid bill. Now, she is in a private room because her health requires it, if she stays in a room with another sick person her health will become worse because her immune system is not functioning properly. Our bill got ridiculously high which a normal family can't possibly pay right away. We have every intention of paying but it will take time or maybe with a payment plan. But St. Lukes will not accept a promissory note or a guarantee letter from PCSO or any Government office. It's either cash or we stay. I'm afraid that my sister's recovery is deteriorating the longer she stays in the hospital. We are planning to just walk out of the hospital but still pay them after. Now, what are the legal implications of this? What case has the hospital against us? I am hoping for some advice since obviously this is not a good idea. But I can't let my sister stay in that hospital any longer, she is depressed and getting worse by the day.