Bleach if fine if used carefully......lots of ventilation and let it dry and air out before putting animals back in there.
First @Sarah99 I would ask why do you need to 'clean'?
Did you have diseased birds?
I hope you don't need to do it now, winter where you are not a good time to be cleaning.
It can also become inert before it kills all the 'bugs'....the more organic material, the quicker it becomes inert.
I would just set a fan up to blow fumes away from birds and to dry it as quickly as possible.