I have 4 cats (spayed & neutered) and one of them, a male loves his rodents. He spends his evenings hunting them. At our old place I would find at least one or two expired ones on the lawn almost every day. This is our first spring in our new home and sure enough, there was one (mole) on the walkway about two weeks ago. As far as what kind of cat, a "Moogie" will do. (mutt in dog language)
I hope you like cats! You can get a loving adult cat desperate for a home at the SPCA in your area.