Are you sure the bully isn't a cockerel? My young lad used to do that to my hens and I did as you did and removed him for a while, and he grew out of it. If it really is a hen then yes, I agree cull her. You can check if there is foam from the vent - if so it's a cockerel.
It is possible there are mites as Harmony says, but a couple of months not cleaning with only 6 hens is not going to cause that.
Can I ask why your cockerel is not in with the hens? I have always kept mine together
All you can do after that is ensure your family do exactly as you ask them to until you are able to do it yourself, and report back with any problems whatsoever - it is a matter of animal health. Teenagers need to learn that lesson
I only have one hen and one cockerel now (the other hens died over a period of time from an illness - they were all bought in), but have the same set up as yours - in a guinea pig cage, in my shed, I have a cover to put over them in winter and a frost free radiator.