My mother is in rehab for broken hip, has some dementia, lived alone with family help. She is failing in rehab, staff wants to discharge.

If she has hit a plateau or she can't do the therapy, yes they can discharge. Medicare determines the discharge not the facility. Your option is putting her in LTC, paying privately or applying for Medicaid.
Breaking a hip for the elderly is very serious. Dementia would not help.

I agree, talk to the SW and tell her there is no one to care for Mom at home.

I would ask them where she is going, since she can not live alone? Talk to the social worker, and make it clear there is no one to assist her.

Correction.....She cannot walk or get to the bathroom.