You need to make a decision first: Is that still your dad? Simple as that (yeah I know!). If it is your Dad, then worry about Mom.

If it's not your Dad due to decline in mental capacity, take the steps to protect him and your Mom:
- See lawyer advice about power of attorney
- Seek Medical advice about in home care, or a home, depending on things. You mentioned you Mom wants him in a home, that right there should tell you where you should start.

Realize no matter what you do, the short term flash-point is going to be huge. But the longer this drags, the worse it will get, if he is suffering from some type of mental decline, it WILL NOT get better and it sounds like your Mom has reached the point where she's asking for help.