Under similar conditions, I usually get a damp washcloth and put it on whatever part of my torso isn't laying against the bed. It's not the perfect ideal (if it does cool down significantly overnight, you can end up *too* cold), but it's enough to help, and is pretty safe for a little kid, too.

I have near-identical thoughts ("how did people LIVE like this") when walking to the grocery store in Wisconsin, in January.