I had a mouse problem years ago in an old house I was living in. We had 3 cats but the mice crawled behind the baseboard heaters and the cats couldn't get them.
I got a couple traps and baited with, well cheese of course, and place one in the basement on top of a circuit breaker panel that was very near the place I figured the mice were moving between the newer section and the older section. I didn't even make it to the top of the basement stairs when I heard a trap snap.
Removed it a few minutes later then reset it and when I came down to check 15 minutes later I had another dead mouse. This continued for just over a day and from then on there were no more mice. I think the total was about 8.
So, look for a place the little bastards are likely to use as a path and they will kill themselves.
Brian