You don't need to rinse off the pyrethrum. It works on contact to kill insects, but it is completely harmless to anything with vertebrae. It's basically an extract from the chrysanthemum flower that acts as a neurotoxin to anything with an exoskeleton.
Mites are most commonly an issue when the...
Most stores with a garden section will have permethrin sprays/dust. It's effective bug control for organic gardeners. DE can cause serious damage to the respiratory systems and should never be used as a treatment for mites.
When one has mites, they all have mites. Treating one bird will not do the job. You've got to treat every single bird in the flock. Then you've got to clean out all the bedding and spray down the whole coop with pyrethrum, paying special attention to the little nooks and crannies. Then repeat...