No, sisters are not legally responsible for one another. So, assuming that they were not already living together, one sister does not have to take in the other sister when she is discharged from the hospital. The hospital is responsible for finding an appropriate placement for its patients and it isn't likely that dumping a patient on an elderly sister who cannot care for her would be considered appropriate. The sister should feel comfortable pushing back.

It's impossible to know where the patient might go if not to her sister's house. If there is nowhere else, the hospital may place her in a nursing home, which may or may not be what she wants. This might compel her sister to take her in. The sister has no legal requirement to do so, but she may feel a moral one depending on the alternatives.