Never had to deal with this problem, but I can say I do use watered down white vinegar or straight up vinegar to routinely clean the cage. For the deep cleanse I use Napisan spray (mix of bleach, clorexidine and other stuff that is approved for house hygiene for babies), let it dry and finally wipe the residue off with a damp cloth. When it’s dry again I’ll re-arrange the cage and put it back where it belongs.

I don’t think mites would attach to any bleached area (unless you live somewhere that’s badly infested, which I hope not for your own safety!), so don’t worry. Be very careful not to do the deep cleansing around the bun’s food or any other fabric or paper/cardboard item that is intended for the bun’s comfort or fun. Also use gloves, cause the stuff will irritate your skin if you use it often!

I believe the method suggested will be both safe and effective. I work in a biology lab and this is exactly (minus the detergent) how we sanitize glassware after we’ve had bacterial cultures growing in it. That concentration of bleach is universally biocidal (kills everything from bacteria, to viruses to microbes like mites). Just be sure to leave it in contact for 5 minutes so it can act. And when you rinse it off and let it dry fully, it does not leave any dangerous residue so it’s safe. You might want to let it dry indoors so that no mites can hop on board as it’s drying.

But, in addition to your cleaning efforts, you might want to look at the bunny’s diet. It’s possible that the wrong diet might be weakening their immune system or maybe they’re allergic to something in their environment. I’ve heard that people with problems like this have seen amazing improvements after switching to a higher quality pellet food, in addition to feeding good hay. You might try slowly switching to Sherwood pellets and see if they improve. Amazon has them and they’re a little pricey but they’re the highest quality feed you can buy (as far as I know). Just my 2 cents 

I had to do a rotation as a shelter attendant during my army career. The best way to sterilize a kennel is to use 1 part bleach with 3 parts water. It’s a rabbit not a tiger, the reason your vet wants it aired out is to prevent respitory damage by the strong chemical small. After an hour or two wipe with a non used white towel to remove residual chemical. Then treat with straight white vinegar or a 50/50 solution, and let sit for a few minutes and wipe with another unused towel. Ensure that you wipe the bars, and change out old willow sticks or toys (boil plastics like baby keys) and wash any blankets or mats. Super important to seriously clean hay racks or litter boxes, if you have any hay bales you get hay from i would suggest new ones or if it’s repackaged hay it’s the same thing.

So checklist:
1. Clean cage with bleach solution 1part/3part
2. Air out for a few hours and wipe with unused white towel
3. Clean out cage again with white vinegar solution to remove chemical smells.
4. Wash dishes blankets, beds and flooring, boil small plastic toys remove old hay water and willow sticks (tunnels are ok)
5. Consider replacing old hay with new fresh hay as there might be mold or bacteria in the hay.