We didn't realize it but Ikea turns out to be the best place to take little kids to play! And the food is very affordable. There's a place at the entrance where they'll watch your kid for free while you shop, but Billy is only 32" tall and it requires 37 inches. I guess it's for 3 year olds.
We all had a blast playing on all the beds, tables and children's toys. It's always hard to find a local place that is free (and indoors when it's over 90 out), and this was a great place to go! Thanks for your hospitality, Ikea!