Hi
Congratulations on having a broody hen. If you decide to hatch eggs, it is important to remember that about 50% that hatch will be males and it is important to have a plan for that situation before you start the incubation process. I butcher and eat my surplus cockerels but some people can't face that. Selling them can be difficult unless they are rare breeds and even giving them away free to a good home can be difficult. Keeping surplus cockerels is also not a good idea unless you are prepared to build them a separate coop once they become adolescent as they will cause stress in the flock.
All that said, hatching with broody hens is a wonderful experience. She is not able to count the days, so don't worry that it is too late to give her eggs. Most broody hens will sit for 6 or 7 weeks waiting for chicks to hatch if you don't break them. That is unfair to the hen though as they lose condition in the process and can cause health issues. If you give her some fertile eggs in the next few days she should manage fine. If she is in a communal nest where other hens can lay eggs, it is important to mark the fertile eggs and remove any others that are laid in the nest on a daily basis but better to move the broody into her own little pen in the hen house so that she is not disturbed. Unfortunately moving a broody hen is not always successful as they bond to the nest site, so if you are going to try it, move her several days before you give her the hatching eggs to see if she will accept the move..... giving her some ordinary eggs may help her settle. She is programmed mentally to return to her chosen nest site, so it may take her a few days to adjust to the new site if you do move her and it is best to fasten her in with food and water until she gets used to it, but some will not settle and get stressed unless you let them back to "their" nest.
As regards the climate, she will be able to incubate the eggs even in winter and by the time the chicks hatch, the temperatures should be improving and she will keep the chicks warm in the same way that she has kept the eggs warm with body heat.... that is why she has plucked her feathers.
Good luck whatever you decide to do.