I'm not comfortable using that because I read it is fatal to cats and we have a cat who likes to hangout with out chickens.
Far more expensive, you can also use Elector PSP. A little goes a long way, something like 9mls in a gallon, which is a little less than 2 TEASPOONS per gallon. It takes 2-3 days to actually kill the mites, but it eliminates all life stages in one go, and has no egg withdrawal period. I'll leave the research for you on whether spinosad (active ingredient in Elector PSP) is compatible with your cat.
Something has to be done to eliminate the mites. Left long enough, they will wreck your hens' health- all of them, very much like leaving a flea infested cat without treatment. Too many discover the problem in much later stages - many threads about that. Mites are especially nasty because the find places to hide at night and attack your chickens on the roost every night.
On the link I sent there were a list of 5 products. I don't include the dog shampoo because it doesn't stop the mite infestation, just temporary relief.
1. Elector PSP (spinosad)
2. Davis Pure Planet Poultry Spray (clove and cottonseed oil)
The remainder are some varying strength/forms of Permethrin.
3. Garden & Poultry Dust with Permethrin
4. Prozap Garden & Poultry Dust
5. Pyethrum Dust