We are putting my mom in an Assistant Living facility, and she owns her own home. She is on Medicare and Medicaid. We want to know what to do with her

You are correct.  Selling her home will most likely make her ineligible for Medicaid because she can only have $999.99 in available assets.  She can rent it, but the rental income would be considered her personal income and would most likely raise her monthly spend-down amount by exactly the amount of the rent.  It is possible for her just to transfer the house outright, but if it is likely that she would need nursing care in the next five years and she was depending on Medicaid to pay for her skilled nursing care, then she would have to deal with a transfer penalty for transferring the asset within the five-year look-back period which would make her ineligible for Medicaid to pay for her skilled nursing care for a number of months based upon the value of the transfer. There are certainly solutions to your problem, but there is not one "right" answer as everyone has different priorities.  I wouldn't do anything before consulting with an elder law attorney.