Why aren't native predators in Australia able to handle the mice problem?
The mice aren't invading Australia, they are invading the wheat producing zones, where the tractors are leveling the land and putting on pesticides. The local frog, amphibian and snake populations downstream of the farmer's fields is strongly affected by the farmland, and so are the birds of prey, if they weather the pesticide use, they have to hide in dense trees, hollow logs, caves, and old barns, which are very lacking in Australian farmland.
If the government enforced laws for field edge habitats for local ecology, then there would be less mice.
Have a look at the map of wheat producing areas.
Rodent Outbreaks: Ecology and Impacts pages 225-238
PDF Google book
The map of mice infestations:


From pestsmart.org.au, though no longer available.

and a map of Australian climate:


wikimedia