If it were me I would let her go broody but I would set something up in the coop where she is comfortable. A cat carrier would work if the space is tiny. You are right it is safer in there. I suspect the eggs are no longer viable if they have been left in the cold for a couple of days. I would throw them out and let her start over or fix up a nest with other chicken's eggs so she has some right away. I guess I should have asked first--do you have a rooster? If not this is all a moot discussion. If you don't have a rooster and you don't want her to hatch eggs I'd look on the forum on how to break a broody.
Let us know how you get on.