Is she difficult to catch? Can you get near the chicks? If so pop them the cage, she's bound to follow? Or if she goes in the cage alone I expect the chicks will follow her. My bantams are extremely tame so I have an easier time when moving new families about. Or another idea, wait until she is...
The idea of the dog cage is great. This will allow the other hens to get used to the chicks, making it easier to introduce them when they're bigger. as they can't squeeze through the gaps. The older hens wont think twice about pecking through a chick head that happens to pop through a gap...
She won't mind being moved at all now... As long as she has her babies that's all that will matter to her. Now they're dry and fluffy and Mobil they're stronger, so will be fine, just scoop her up, place her on a nice clean nest, with chicks crumbs and shallow water near by in a nice safe place...
I'd move her to a brooder or somewhere quiet. I'm sure she's coping, if she's managed to hatch them all they must all fit under her, so at least they'll be warm. We're they a surprise hatch? Exciting! :P