So, the summer is over. I just wanted to update everyone. I just used scotch tape to affix the mylar over the individual panes, inside the room. (am I using the right term? there's just two per window, they slide up and down to open).
It really helped a lot, I think. I don't have hard numbers as evidence or anything, but it was just much darker and cooler in the room all summer long. I would definitely do it again. I didn't really get to the cardboard because I was anxious to get the stuff in the windows, and finding and cutting cardboard was just another step. The mylar came down quite easily and is sitting in the hallway, waiting to be folded up for next summer. I do think a kid took off with one of them to play, and then DH trashed it. Oh, and I used one of them to wrap a birthday gift for a kid's party over the weekend. Oh well. I only used 1 or 2 sheets, and that was out of ten, so it's not a big deal.
Maybe next year I will try to put it on the outside. It was just so much easier to just tape it onto the inside.