I wouldn't use bleach, instead I would use oxine if you want to disinfect it. It will not hurt the birds if they breath it unlike bleach. Here is their brief description and the link that has the full write up follows. I doubt that it will kill mites, you will need something like pyrethrin to do that.
"Oxine is known to kill every bacteria, virus, and mold it has ever been tested against and is 200 times more effective than chlorine bleach. But one of the most impressive things about Oxine for me is that it does it with such relative safety (when used according to label instructions). Environmentally speaking, Oxine actually biodegrades to ordinary table salt. And it is so safe to use on livestock that it is actually approved for use in the drinking water of organically grown animals"