I'm afraid I think your young pullet may be sick and is seeking sanctuary in the nest box rather than wanting to brood. This occurred to me, as well.
You say that she is unsteady on her feet and that rings alarm bells with me as she is the right age to be vulnerable to Marek's disease.She did eat medicated feed til she started laying. Isn't the medicated for Mareks? (sorry these are my first chickens) Another possibility is that she is egg bound although you have checked for that. Has she actually started laying eggs yet and if so, how many? She has been laying beautiful blue eggs for about three months, about 3 a week I'd estimate. It's unlikely she would go broody if she has only laid one or two eggs or hasn't actually started yet. It's also not a good time of year if you are in the northern hemisphere and a novice broody hen of that tender an age is unlikely to be very successful in winter, so might be worth breaking her and hoping that she will become broody at a more appropriate time of year
Another possibility is that she has been kept from the food and water for some time by a bully hen and her system is shutting down,,,,I have three feeding stations for 11 chickens, so they all get to eat. I also scatter food and treats so they don't have to compete for food. that would also make her wobbly on her feet. Has she pooped and if so, what is it like. Broody poops are huge disgustingly smelly dollops. If you get one of those I will be very pleased to hear it, but my concern is that you will get a thin watery poop, possibly white or green. Her poops have been scant, green, with a little white at the edges.
I've had hens that were at deaths door fight me with amazing strength when trying to feed and treat them, so don't let that in itself fool you into thinking she is well. Check her condition. Can you feel her keel bone and is it sharp or reasonably well covered? If sharp, that's another indication that she's ill rather than just broody. I will have to check on that. Her crop is empty. But I'll have to find a diagram so I know where to feel for a keel bone.
Really hope I'm wrong, but there's something in your post that just doesn't sound right to me.
Good luck with her either way.
Watching a broody hen raise chicks is the best chicken entertainment there is, so I hope you get to experience it.