Getting a cat does not work!!
I treat sometimes 5 houses a day for mouse infestations about 50% of these people have cats.
The problem with cats is they usually start the problem in the first place, they catch mice outdoors and then bring them in alive to play with them, its unfortunate that most catflaps are in the kitchen door, so the cat brings the mouse in doors put it on the floor to play with it and the mouse shoots off under the fridge/cooker/cupboard etc, now if this mouse is pregnant you have the beginnings of a big problem.
Mouse traps are OK but you need to put down a few and keep monitoring them, they are best baited with peanut butter, they love it and cant run away with it, which happens with cheese, chocolate etc. nothing is as effective as well placed mouse poison!!
Yes it does happen that on occasions you will get a mouse die under a floor, this however does not happen very often.
Good luck