Congrats on the broody! You say she already has her eggs, and you didn't mention her being bothered by other hens in the coop or being chased out of her nest... so I would hesitate to move her at this point. It may upset her and cause her to pace and try to get back to the original nest, meanwhile her eggs would possibly be neglected too long.
If you do need to move her because of problems with other flock members... I would do it at night, have the new box ready with some of her old bedding, just use a head lamp to minimize light and move her and leave her new area undisturbed for a day.
If you think it is ok for her to stay where she is then let her go till after she hatches, then move her and the babies to a ground level coop/box. Though if the coop is big enough to allow her to move around inside with the babies for a couple of days she would probably do fine where she is and she will take the babies out into the run when she is ready...but you will have to be very vigilant that the babies don't get left behind in the run when she returns to the coop, since they take a couple of tries to get used to using a ramp.
Edit to add.... 2x2x3 or 4ft long should be fine for a broody hut for a few days. It can even be a wire cage with a blanket or tarp over it if need be and the weather isn't too cold. Though I would suggest the nest end of it be solid wood of some type with a pop door sized opening on it so she has a snug, draft free nest area. I have used medium dog or cat carriers as nesting boxes with good luck, a sturdy cardboard box could also be adapted for single time use as long as you can cover the cage area to keep it dry.
If you have a hardware or construction supply company nearby you could stop and check to see if they have boxes from fridges or freezers that you could get one.... another box which is perfect size is the ones they package the truck bed toolboxes in... so a garage that does custom installation work like that may have them. With an oversized box you could easily partition off an end for the nest area, and the rest of the box you could cut holes or 'windows' in up high on the size to provide lots of light and cover the windows with hardware cloth for safety. create a pop door at the end farthest from the nest with a wire door you can open or close as needed.
Using cardboard box can only be done if you are sure your run area is secure from predators, the cheeping of the chicks is going to draw a lot of attention from unwanted critters.. so security is a must.