Don't do the cold bath thing this time of year. Even in summer, I don't like the idea. I tried that with a couple of mine before and it didn't help anyway. Just made the hens not want to be handled anymore. My hens had already hatched several clutches, and it was getting late in the year or I would have just put some more eggs under them.
I put mine in a separate pen with food and water, and a roost, but no boxes or anything to get behind or into to make a nest. It only took a few days before she was acting normal again.
A wire bottom cage is good, but only in a heated area, so she won't get too cold. Lower body temp will help, but not to the point of letting her get chilled. You don't want to kill your hen!
She does sound like she may be an Australorp. Maybe later you could trade her for a breed that doesn't go broody, if you don't want more chickens. You could be fighting broodiness in her pretty often. You can't change her basic instincts or hereditary hormone cycles. Maybe you can find somebody with non-broodies who wants to hatch chicks. I see posts all the time from folks wanting to make a hen brood, or get one to stop. Better to just get the breeds that are right for you, that naturally do what you want them for.