My 90-year old mother has been going through the same thing for the past few months. She had dementia and has been sent to the E.R. from her rest home a number of times, ranging from bronchitis to kidney failure, and they absolutely REFUSE to refer her to skilled nursing. Same with my BF's mother, but she finally died while waiting.
Welcome to our Brave New World.
And BTW, my mother has excellent private insurance in addition to Medicare. Medicare simply REFUSES to take the assignment to a skilled nursing facility, where she needs to spend some time.
It's a nightmare, folks.