My Mom is currently in a nursing home in NY for rehab. She would like to remain in the nursing home but they are trying to force in her into assisted

There are screening tools called Patient Review Instrument (PRI) that determine whether someone needs nursing home level care.  You could possibly hire your own nurse to make the assessment. Nursing homes often do not want residents that require little care, because the patient mix determines their overall Medicaid reimbursement rate. New York does have some assisted living facilities that are covered by Medicaid. They are part of the Assisted Living Program (ALP).