Can my wife give one of the kids power of attorney if we are still married and not having problems if she is on her death bed

Sure she can. What is your worry about someone else having a POA ?

Yes, POA of course ends at her death, so on her death bed would be a silly place to give it.

As said of course she can. She is free to choose whom she wants to handle her business. But as said POA ends when the person dies.