Our mother has been moving between home and nursing home despite fantastic home care situation. She really would be excellent candidate for assisted living as she has mobility issues but is very sharp and otherwise has any other health issues well managed. She cannot afford financially to pay for both her apartment and the nursing home but seems oblivious to that thinking they can "put it on payments. Please help us understand how to deal with this and why she would be acting this way.
I believe that a reputable rehab facility will not keep working on somebody who has no need and is not making progress. Maybe the ER doctor thought 3 days would be enough but other things came to light once she got there. Have you talked to the therapists or do you have only your mother's version? If she tends to be a drama queen it is easy to dismiss her claims when they sound attention-getting, but sometimes even drama queens have dramatic problems.
Once she makes a decision, and only then, you can help her face the financial realities. How many years will she be able to afford to self-pay for the NH (or an Assisted Living place)? Presumably if she outlives her funds then Medicaid is in her future. But don't go into any of that until she decides what is the best place for her on other grounds.
Good luck!
This is a puzzle. Give a few more pieces and maybe someone will have some suggestions.