I think sometimes as pet owners we forget that these are natural behaviors that would be exhibited if the cat were in the wild. Regardless though, with children involved it’s obviously important to maintain a healthy environment so there are a number of things you can do to prevent this from occurring.
One thing is to put a bell around the cat’s neck. While cat’s frequently use the crouch and wait attack method, it’s still more likely to give mice a chance to escape than not wearing one in the first place.
Another option is to keep a cat indoors. This isn’t an option in all circumstances but it’s an option for some and may be the best 100% guarantee if it’s really becoming a big problem.
One other way if you live on a rural property is to put traps and snares out for the mice in areas where you know mice frequent like barns, animal food storage and other areas.
To ensure these wild animals don’t enter your home – yet still allow the cat outdoors it may become necessary to remove the cat door so they can’t bring them into the house itself.
It may be possible to wear out the cat so it has no energy to continue this behavior, but in all likelihood even a well fed and well entertained cat will still hunt.