I have had 7/8 Black Rock hybrids for 3 years now. I don't know about lice but I have a little bit of experience with red mite but I'm not an expert. Do you know that the lack of eggs is mite-related? It might be something else - age, rats.
If it is mite-related then I would imagine the mite infestation is serious. It is really the hen house that you have to tackle. The mites will be living in all the dark nooks and crannies. Cleaning them out is the worst, most skin-crawling job ever. Tie your hair up in a headscarf, wear long sleeves/trousers and arm yourself with a torch and look really closely at the walls, roof, windows, door hinges, roost bars etc in the henhouse.
Essentially you have to sweep every surface and all the joins and corners. Then sweep up all the bedding and dispose of it. Spray a permethrin-based insecticide according to the can. Leave the house to air for a few hours before adding fresh bedding. Then go inside and run a bath and pour yourself a gin & tonic.
Earlier this summer I creosoted the outside of my henhouse and was quite impressed by the mite carnage that resulted. So when we had a dry day recently I took the roof off. There was quite a mite population living in the gap between the roof sections and walls. I scrubbed with a wire brush, let the sun in and then painted the tops of the walls and the worst bits of the inside of the roof with creosote. I have made new brackets for the roost bars which I have creosoted too. I just haven't got 'round to fitting them.
I suspect that the creosote isn't too good for the hens but we leave the pop hole open for ventilation and they are only in 9pm-5am at this time of year.
By the way although I have had a wee problem with mite the girls are still laying, so I'm not sure if mites is what is causing your problem.