I finally decided to finish the roof on my run. I had it half done in 2x4 welded wire but didn't like the look at all, and couldn't motivate myself to finish something I didn't like. The Easter Egger girls could launch straight up and out now, so yesterday I talked to my wife about corrugated steel vs lattice, and we (she) decided on lattice. Off to Lowe's yesterday afternoon, and I redid the whole thing in about 2 hours. I LOVE it, and my wife is happy, so that's great. I think it turned out way better than corrugated steel would have, and it won't hold snow as bad.
We may plant some grape vines around the outside to help offer some additional shade (and perhaps drop some tasty morsels to the girls) in the summer.