Good idea!Since there are so many of them, maybe you (or the kid) can take a piece of tape and start catching the mites from the crate. Just stick the tape to the mites, continue until the tape is full, then throw the tape away. You could put mite-covered tape into a tightly-closed bag or jar before throwing it away, if you want to be extra-sure of those particular mites staying gone (especially if you already have an empty jar that needs throwing out.)
I would not expect to get ALL the mites that way, but catching a bunch of the visible ones might make you (or the kid) feel like something is being done.
@Farmerbetsy I would be sort of freaked out too, I think anybody would be.
If she's in sad shape, moving her to a colder room is going to be harder on her. Do you have a way to safely offer her some heat if you did move her? Space heater, heating pad, heat lamp, etc.?