Nursing home wants private pay, while resident (who is already admitted) lists house for sale for money. Resident can't apply for Medicaid, until house is sold (it has a mortgage) and money is spent down. No other funds available- Nursing home says POA has to pay or resident will be discharged. There's no one to care or take in resident.
You can own a home and still apply for Medicaid. Who told you that you have to wait until it is sold?
What other assets need to be spent down? Is there enough to private-pay while waiting for Medicaid?
Unless you signed something making you responsible you do not need to pay the nursing home out of your own funds. But they don't have to keep the resident indefinitely without getting paid. How long has the resident been there?
How certain are you that the house will sell reasonably soon? How confident are you about the price? How many months of NH costs can be paid with the equity you expect from the sale?
I wonder if it might be worthwhile to consult an elder law attorney to get you started on the Medicaid application process and advise you on options for dealing with the NH in the meanwhile.