My parents have a mouse problem (in their barn) so they borrrowed my cat! She's been there since mid summer and has just about killed everything there is to kill.
But, in the past when mice were found in the house, they used the poison that kills the mice then 'dehydrates' them so they don't stink. We were very carefull about where we put the poison to keep the cats safe and found that removing the kickboard under your kitchen cupboards or under the drawer below your stove are decent places. You can go the trap route, but its going to take that much longer... Its not the greatest, but I'd go poison.