Hi
Firstly it is important to understand the difference between lice and mites and what to look for to confirm their presence before you start throwing chemicals at them.
Red mites live in the cracks and crevices of the coop and crawl onto the chickens at night to suck their blood. Generally they become a problem in the warmer weather and are not an issue during the winter if you have a cold climate. To check for mites, use a soft brush to flick debris out of the crevices of the coop, particularly those nearest and under the roost, into a dustpan and then tip onto a piece of white card. If you have a mite infestation you will see small round red specks which when squashed with your finger nail, leave a smear of blood on the paper.
Lice live on the chicken and are grey/light brown. They can be difficult to see but what is easier to notice is clumps of their grey eggs clustered at the base of the chicken's feather shafts particularly near the vent and under the wings. They live off skin and feather debris and are more of an irritant than a health risk. Usually chickens will dust bath to rid themselves of lice but if conditions are very wet or they don't have access to a dust bathing area, then an infestation can build up and make their lives uncomfortable. Usually lice will be found on a sick chicken that is not looking after herself and dust bathing properly.
Lifting birds down off the roost at night and using a head torch to see, parting their feathers and checking the vent and under wing area is usually the best way to see them.
Then there are scaly leg mites which obviously aren't the problem in this case.
And depluming mites which I don't know a lot about as I've never had to deal with them. They are less common and I believe are not responsive to poultry dust and may need a systemic pesticide to treat.
Chickens normally attack sick birds, rather than birds that are just suffering from mites or lice but as previously mentioned, sick birds will often have problems with lice as a result of being sick.
I would thoroughly examine your hen for other signs of illness. Check that her crop is emptying overnight. Monitor food and water intake and her poop. Make sure there are no unusual swellings, particularly abdominal swelling and of course, check for lice and mites. Then decide which chemicals might be appropriate to use.
That would be my advice
Regards
Barbara
PS. Unnecessarily treating a sick chicken for lice can at the least result in extra stress and strain on it's system. I had a sick chicken die shortly after dusting her with poultry dust just because she was weak and the stress of it killed her. She did have lice but she didn't need manhandling like that when she was already so ill. I felt really bad and wouldn't want anyone else to experience that if it wasn't necessary.