Please feed your cat! Just like some hens will go broody and some won't, good mousing ability is not what today's cats are bred for.
I find that the mice in my house that get past the cats and dog (GREAT mouser) are smart enough to stay within the walls and ceiling, also crawl spaces, etc. So the cats and dog will never get to them.
What works well here is to put several snap traps inside a sturdy shoe box or small cardboard box. I cut several mouse-size doors in the box and bait the traps with peanut butter, just a dab under the trigger. Put the box where you have seen or heard mouse evidence and be patient.
I like using a box because, depending on how the mouse is positioned when it triggers the spring, it can be a bit messy. Or an incomplete kill and the trap gets dragged away....you don't want this to happen, as you might not find it, especially in a crawl space.
We live in an older home with lots of established tunnels in the walls, etc, so every fall we are battling unwanted tenants!
You can also buy snap traps made of plastic that you can re-use, but if you are squeamish, the cheap wooden ones can be tossed in the trash, mouse and all.