First thing to do is take her out of the nest box and check thoroughly for red mites, a broody is literally a sitting target for the horrible things! They look like tiny little red pinprick things and there may be tiny white bits around the area. You may need to lift perches out to check.
If there are, make a solution of dettol (the Orange/brown, strong smelling stuff, not the kitchen spray) and spray it into all nooks and crannies of the house, once it's been cleaned out. This will need to be done every two days for two weeks (they have a similar life cycle to headlice). The dettol breaks down their waxy outer layer so is a very effective anti-mite tool.
If no mites, then take her out and have a really good look at her, is she very thin? Does she have lice on her? If you can encourage her to eat, cat food (tinned meat) can be a good tonic for a broody hen.
IME vets don't often know much about chickens, unless you're lucky enough to have an avian vet nearby, but it might be worth ringing and seeing what they say.
As for chicks, if you can find a box to put them in, take them away from her (if she doesn't perk up). You'll need a heat lamp, even a light bulb will do the trick, in a corner so they can get away from the heat. If you have a country store or a good pet shop near, they will probably sell chick crumbs and other equipment. If you can't immediately find chick crumbs, some finely mashed hard boiled egg will do for a day or so - put some on the floor and tap your finger near it, they will come and peck around. They will also need water in a shallow dish, dip their beaks in if they're not sure what to do with it.
If the hen perks up quickly, the chicks can go back under her within 24 hours, after that it's more likely that she will reject them.