What happens to POA if competent principal (Father) is in hospital and agent (Son) dies?

If the daughter is secondary on the POA then yes she is next in line but, a lawyer may need to put that into effect. Then, how does the POA read, if there is a need for a doctor or more to say principle is incompetent she has no say until that is done. Immediate, he still needs to be incompetent.

The POA has to say she has the right to sell. Just because she is the daughter of the POA does not make her automatically the POA.

I am having a hard time interpreting what you say here. Best I can understand is that you say that the principal is in the hospital.
If he is not competent and the POA has been acting for him, and the POA/Agent has died then the document needs to be examined for a "second". The second listed on the POA would become the acting agent automatically. It is highly unusual that there is not a second mentioned.

If the dead POA was a son and there is a daughter, then it will be an easy thing, if the principal is deemed not competent.to make his own decisions, for her to be appointed emergency guardian/conservator. She would then do placement and sale of home, yes.

May I ask what YOUR concern is in all of this?

Normally you would have a back up POA just in case. I am the primary and bro is secondary for momma. Both for medically and financial.

This is the reason people usually name more than one person to act as POA, sometimes to act jointly, sometimes jointly and severally or sometimes as alternates. It's possible the daughter might have been named as secondary but it doesn't just get passed on to her like an inheritance, as olddude says you need to see the document. And whoever they are a POA acts as an agent for the principal, they can not override the desires of a mentally competent person.

You need to read the document to see who the contingent POA is. The daughter may be right. Or maybe there is no contingent POA. POA agreements are not all the same.