We have a couple of foxes that come around every night, since they arrived on the scene I've had very little rat problems. Someone told me they take up to twenty rats per night.
As for your mouse problem, set the traps and check them every couple of hours. I use peanut butter on the ones in the sheds and carefully set it that it will go off with the slightest pressure. A friend of mine has a business and has the back room covered in all sorts of traps and electronic high pitch sound emitters but because he doesn't really make a proper effort the mice are still there and the traps are lying all over the shop. If he only put his efforts into one or two traps rather than saturating the place every so often and forgetting about them.