If the coop itself is that infested, you should probably get an insecticidal bomb to rid it entirely of bugs. (after of ourse you get rid of all litter and clean it thoroughly) Read the label for how long it should be left empty, and make sure it is at least that long. Longer would not hurt. You will of course need somewhere secure for your birds during this timeframe.
You should also treat them. Depending on the number, you can bathe them in flea/tick shampoo, spray with flea/tick spray or dust them. Personally I prefer bathing them, and I have LOTS of birds. At this time of year getting them chilled is rarely an issue.
Ivermectin pour-on applied to their skin (I prefer the wing pit; some use the back of the neck) is effective against pests that bite the birds; I don't think it will affect feather mites, however. It is also a broad-spectrum wormer.