I would not use Seven on the broody or her nest. The mites will not harm the chicks that much. If I were you, I would get some DE and use that to treat the broody and her nest. You can even put it on the eggs, it will not harm them.
As for treating the birds and then the coop, that should be OK if you do it the next day or very soon. Bleach water is not effective in killing mites and in my professional opinion would be a waste of time & bleach (I work in the pest control industry).
There are different types of mites; poultry mites, scaly mites and feather mites. Poultry mites do not live on the birds. They move onto the birds to feed and then off again to a harborage. Very similar to bed bugs. Feather mites live on the birds, usually at the base of the feathers. Scaly mites also live on the birds typically on the feet & legs, but if bad enough they can also be found around the beaks. The treatment is very similar for all. Treating for mites that live on the birds can be done with soap and water and is very effective, but it takes numerous washings and if you have very many birds, it is a lot of work. Feather mites & scaly mites are normally passed to other birds by common nest sharing and from hen to babies. Poultry mites require a total coop cleaning. Since no animal safe chemical will kill the mite eggs you must retreat in 7-10 days to kill any mites that hatched since the first treatment.
If you are going to treat the entire coop, I recommend that you remove all bedding and treat with DE and/or Seven. Do not replace the bedding. Re-treat in 7-10 days and then wait a couple more day before replacing the bedding. You can put bedding in the nest boxes but it should be removed and destroyed prior to the follow up treatment.
Seven is a good pesticide and it is very durable especially when not exposed to the sun & rain. Do not clean it up before replacing the bedding. Leave it in place and it will give your birds residual protection. Also, pay particular attention to where your birds roost. Use a mask or respirator when spreading any dust in a confined space. Also keep your birds out of the coop for an hour or two after spreading the dust.
As a preventative, I use DE & wood ash in a dust box, especially in the winter. Contrary to what some believe, DE is a very effective bug killer when used properly. Wood ash is good because of it's consistency. It can penetrate into the scales on the legs and to the base of the feather and mites do not like it.
I hope this helps and good luck.