If mom goes to a NH and signs away her assets what happens if they aren't all used?

Signing them away usually means they are now the property of the Nursing Home. If mom dies, any unused funds are returned. Generally the house is sold by the Nursing Home and it does not cover all the bills. Let's say you are lucky, the house sells for $100,000 and the NH bill plus the commission of sale and the closing cost is only $50,000. The extra 50K would be sent via check to the The Estate of the deceased. It never happens.