I have been pet-free for about 5 years now and enjoy not having the added expense and trouble. But I've had cats (and several dogs) ever since 1974 prior to that. Cats will certainly help with a mouse problem, but it somewhat depends on the specific cat. Living in the city, our bored indoor cats liked to play with the mice. The female would even carry them around like a kitten. After awhile the mice would die from this kind of abuse.
Then with an old cat, they would eventually "retire" from mouse-catching. They would make some kind of effort but not all that good. Regardless, I think the mice could smell the cat and they would keep them away to a certain extent. My last cat was like this when he died at age 17. Funny thing is, just a couple days after he died mice got into the kitchen and made a big mess.
Of course, the mice can also get into drawers and cabinets through routes not available to a cat, so you will still have some issues. These days I just use the grocery store variety of mouse poison and it has been very effective. I find this is essential out in my toolshed. If I don't keep the poison current out there the mice build nests inside the engines of my snowthrower, lawn mower and generator. They ate through some internal wires in the generator and that was a pain to fix. Even with the poison, this winter I fired up the snowthrower to test it (never needed to actually use it this winter). When it wouldn't start, I opened it up and found a nest with a dried out dead mouse (maybe a rat, it was pretty big) in there!
I'm not so keen about using poison but I just don't know of any other practical way to control the mice here. In one evening they can ruin all your packaged foods and leave droppings all over everything in your drawers.
