My mom wants to move back on her own, but she is a fall risk, has left stove top on, and refuses to do anything. Any advice?

If you have no legal authority for her and she can no longer (or is unwiling) to assign you as PoA, then call APS and report her as a vulnerable adult. Just keep calling and reporting her. Eventually they will move to acquire guardianship and will place her in a facility for her own safety.

Can you tell us where Mom is living now, and why?
I think you already recognize that what Mom wants, and what is the best safe placement for her are two different things.
Does Mom has a diagnosis of dementia?
If not, she is still in charge of her own decisions.
Do you have POA for Mom?
If not it is too late to get that if Mom has dementia, unless she is still capable of understanding and making this sort of decision. An attorney could help the two decide if she is capable of that.
What you can do legally depends on the answers to the above.

Move from where? More details needed.