Cat!
Seriously........
I don't even lime cats but we had a similar problem in a house we bought a cpl years ago. Like rats destroyed the insulation and heat ducts twice in a few months (prior to our purchase, bought a foreclosure that had been unoccupied for a year).
Didn't see one mouse or rat turd after getting a cat.
Just moved to AK. After a cpl weeks of getting the cat acclimated to the new house so he wouldn't run away, we let him out 2 days ago. He's brought back 7 or 8 mice so far and we don't even have a rodent problem here.
Last house w the rat and mice problem he kept our 5 ac and the neighbors 5 ac rodent free. Outbuildings and everyhing.