Baby wipes and dry shampoo. Extra set of any other toiletries that you use (deodorant, etc) kept in your desk. Also ALWAYS keep a clean pair of socks/underwear in your desk drawer.
That's pretty much what I do, also. I keep a towel at work, also...and bring a new washcloth each day in the summer. We do have an enclosed bathroom...not just the standard public-stall-variety. So, I can go in there and cleanup.
I have a miniature tornado fan in my office to blow cool air on me.
I run year round...including July and August here in Dallas. I've done long runs in 105F, and still managed to not scare the entire office away (the weak ones needed to go somewhere else, anyway).