I agree it is time to address your problem. My guess is that she is eating secretly at night or when you are not around, but it is not good for her health to be in a perpetual state of broodiness.
Your girl can hatch a fertilised egg from any breed of hen, but if that's the path you are going to go down, I'd give her more than one egg to ensure something hatches in due course. In giving her just one there is the risk that it won't be a viable egg and you will be no better off in 3 weeks time.
The alternative is to put her in a broody buster cage and break her of her broodiness. This is a wire cage with a fine wire base, which you should put up onto blocks. The only thing that goes in the cage is feed, water and your hen. No bedding what-so-ever.
Lock the hen in the cage (make sure it is protected from the elements of course) and she should be broken within a week or so. You can test her out after 48 hours - let her out of the cage and watch what she does. If she runs back to the flock she is more than likely busted, but if she heads for the nest she needs longer in the buster cage.
Good luck!
Krista
