I had a big problem with mice and rats when I first got my aviary. I used a box trap to catch the mice, but the mice soon disappeared once the rats started getting in. The only problem with that was that the rats ate all my quails and half my finches.
I took drastic action and got all the remaining birds out, and my husband cemented the floor of the aviary, then bolted the aviary down so there were no possible gaps. Haven't had a problem since.
I spread beach sand and shell grit on the floor of the aviary, then every month or so I sweep it up and lay a new layer. I've found the floor has been much cleaner and dryer since cementing it.