I'm curious as to why he appears to be dying. Makes one wonder what he/she has been in. Are you certain it's a baby - not that that makes any difference. Where there is one there are usually more.

We just use the sticky trap and catch them and put them in the trash so that some poor cat outside wouldn't get their feet in that sticky stuff. They seem to find a way (1/4" hole/crack) will let them in. Usually this happens in the fall when they want a warm place to go.

I really don't feel bad for mice as they really can do damage. They will pee and crap on everything and chew stuff up and -- have lots of babies to join them in their escapade. Git rid of the thing - kill it one way or the other and don't touch it with your bare hands.