What does a nursing home do when the POA for a patient dies and there isn't anyone else listed as secondary?

If the person whose POA died is competent to make such decisions, he or she can name a new POA. This would be the most desirable resolution.

My mother was one of the majority 97 mentions -- she had no POA at all. The Nursing Home had no problem dealing with that. If the POA was paying the nursing home they might need to have the money sent directly to them.

I haven't been through it but I'm assuming it would be much the same as anyone else who doesn't have a POA. Many, if not most, people don't.
Usually decisions are deferred to family members. The spouse, the children or brother or sister depending on age etc.
If no POA assigned or living then guardianship can be sought. NH will seek guardianship in some cases. The state laws will prevail. 
Do you have a loved one this has happened to ? 

I am moving your question back to the front of the list. Hopefully a caregiver who has been through this situation will answer.