Dawn dish soap is supposed to be pretty good at killing mites. I used it to bathe some hens that had chicken mites the other day. A number of the mites rinsed off the birds. I think it was mostly from water-saturation/drowning from the long "shower", though.
For a chicken with tiny Northern fowl mites, I have submerged the bird up to the top of its neck in a plastic bin of warm water. Before submerging the bird in the water, wet its neck & the top of its head. Massaging some soap suds into the feathers around the top of the neck is also good. This will help discourage the mites from fleeing up the neck for "dry ground" when you mostly-dunk the bird (I learned this while bathing dogs at a kennel). Keep the bird submerged for a couple minutes. It should relax well while resting in the warm water.
This worked wonderfully for me on the Northern fowl mites--They rinsed off dead in droves when I shower-rinsed the chicken in the tub afterward. I imagine it would also work well with chicken mites.
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If you can avoid it, I wouldn't give your hen a mite-ridding bath when she's weak because of the extra stress, unless you're sure she's swamped with mites. You can check some of her coopmates for mites. If she had them, they will.