But in the meantime I put the query out on Twitter asking for some other good spots to spend some time with a laptop, and I got some answers. Feel free to provide your own in the comments, though.
Coffee-wise, Tynan at 14th and Irving and Flying Fish in Mt. Pleasant were suggested, though Tynan has a no-laptop-before-3pm rule on weekends and also a time limit, so that was out for me. I understand why they have the rule though, as it's a small place and you don't want people hanging out for hours with just a coffee if you can't fit anybody else inside. I'll have to try Flying Fish though.
Sankofa, the bookstore and coffee shop at 2714 Georgia, and the new Coffy Cafe were also suggested. Sankofa opens at 9 on Saturday and 11 on Sunday. There's also Columbia Heights Coffee on 11th, the grandaddy of neighborhood coffee shops, which has free wifi but is quite small, and Sticky Fingers on Park. And Starbucks on 14th, of course, but that place is often mobbed on the weekends. The one at Georgia and Bryant might be an option, though, and Starbucks now has free wifi.
As for restaurants, if you were getting lunch or something you could do work at The Heights or Dulcinea, both of which were mentioned by Twitter folks. I presume you'd have to buy a meal though.
So those are my ideas, did I miss any good spots?