You can get the 'earth friendly' raid in your home and garden center, and use that to kill them. Cover your Chin's cage with a sheet, and spray the bugs. Make sure you wipe down the walls and surfaces really well though, because even though its natural pyrine oil (from the chrysathemum flower) it can still be toxic to animals. Soap and water will kill infects on surfaces too, but with flying ones it's not going to be very effective. One of the things that we do is treat our pellets. You can place them in a pan in the oven at 200 degrees for an hour and it will kill and insects and larvae that may be in your pellets. That is the most common 'point of entry' for insects associated with animals.
If you don't want to use a spray (which is completely understandible) you can use a vaccum with a hose to suck up the bugs you can see, and make sure you clean the cages throughly and remove any trash. Put your food and hay in an airtight container (I know this is not recommended for hay, but if you do it for a week it should still be fine), and change the bedding every other day until you don't notice the bugs anymore. Diligence is going to be your friend!