My hens are all wild so I understand. However, if they're dedicated broodies, you can quickly reach in the nest and grab them. Wearing gloves will make you less apprehensive.
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Methods other than the elevated cage with no bedding have marginal results at best. A warm bottom promotes the hormones for broodiness. Dunking the bottom will not keep the bottom cold enough long enough.
They do have to be in the cage 24/7 or the hormone cycle won't be broken. The dimensions aren't important. A broody hen sits tight so the cage only has to be large enough for her, a food and a water dish. Once she's up wanting to get out of the cage, she's likely done. I pull them out after 2 days. If they go back to the nest, back into the cage she goes.
In poor areas of SE Asia, Africa and Latin America where wire for a cage may not be available, they'll use sticks bound into the shape of a cage.
I've used ice on occasion but nothing works better than an elevated wire bottom cage. As I said, this method has worked for thousands of years.
Chicken keepers over the millennia understood why chickens wanted to raise a family and how to quash that desire. That kind of information passed down over generations of keepers has been interrupted by the bulk of the population moving to urban areas.