Here in the UK, this would be a job for a working terrier. You can hire terriermen with a pack of a few dogs who will come in and sort out your mice or rat problem very fast. I'm not sure if they are widely available in the states, and admittedly lots of dogs going after mice and rats is very gruesome, but it has no poison and is very effective at clearing out an infestation (see examples on YouTube). If there are any in your area, I'd hire them immediately. Your problem would be sorted out in an hour or two.
One of my terriers (just a pet) likes to plant himself outside by the coop and he catches things all day long for fun. He and the cat have a little routine together. Its just instinctual.
There's a YouTube channel called mousetrap Monday (linked below) which shows how the spinning bucket mousetrap is intended to work. He also has lots of other reviews of traps on there that might be of interest. Good luck!
https://www.youtube.com/user/historichunter