When its freezing the red mites wil die. The mite-eggs however, won’t. Depending on the harsnes of you’re climate? Be prepared in spring to check daily when its getting warmer again.
If you keep the old coop (handy if you need to separate a few chickens / a broody / roosters).
Keep the chickens out for long time. Do clean the coop again in spring and use chalk-paint for pigeon-lofts mixed with DE. Cover all the cracks thoroughly. This is where the mite-eggs are hiding. The paint has 2 advantages: you see the mites very easy if they come alive. And the newborn mites dry out on the chalk and are damaged by the DE. It should be safe to use the coop again when a year has passed.