Fun fact, cats aren't actually very effective at killing all the rodents. They've done studies and basically the cats just make them really skiddish and they hide so you never see them. They kill one here or there but the real benefit is getting them out of sight.

That kitty litter solution wouldn't work for mice with toxoplasmosis, plus cat pee doesn't smell particularly nice...

I agree there would be droppings if there are mice.