This is exactly what I did. Grear thing about the dog runs is you can add panels and expand the run. This one went from 10x10 to 10x20. Aibe tbe chain link is sturdy, I went with a nylon type plastic fencing around the bottom on the inside. Wonderful stuff! Cuts with scissors, doesn't tear you all up when working with it. I think being inside it will be harder to chew through.I would have the coop outside the run since you have that many chickens also because if you put put the coop in the run 18" off the ground they are liable to decide to lay eggs under it, making it difficult to gather them up. Put a few obstacles in the run so they have a way of getting away from each other in case they decide to have a chicken fight. Leghorns are not that big so they might be alright for a while in the space you have, it would be nice if they could get out and free range some to stretch their tail feathers.They love to fly so maybe put a chicken swing or places to fly up to in the corners to check out and get a good seat away from others when they need it.
You want the house inside the run, and they will love getting under it, especially in hot days.
I used 2' chicken wire around the bottom outside, laying it out on the ground 1' out and staking it down, then wrap up the side of the run wall. Hoping this keeps varmints from digging in from the outside. We have empty fields on 2 sides. (I did lay boards on top of it, so I don't trip on it)

