Australia is currently dealing with a mouse infestation problem, but Australia is awash with different types of predators that presumably eat lots of mice like animals. Snakes in particular are well suited to kill mice, especially in close quarters. Snakes can fit in places most other predators can't due to their size. Additionally large invertebrates (like huntsman spiders) should also be able to take on some of the mouse load.
And it would be one thing if this was the first time the population ever boomed like this, but mice population booms have apparently have happened since Australia has existed. So why aren't we seeing large spikes in those predators when mice populations boom like this?