The Magic House is a must. There's lots of stuff there that a 2-year-old would be interested in. It's in Kirkwood, which is an inner-ring suburb, and it's a quick few minutes off of Highway 44 (not sure where you're staying, but it's no more than 25 minutes from downtown). The Zoo (in Forest Park) is also awesome. Entrance is free, but spring for admission to the Children's Zoo. There are enough cool/air conditioned parts of the Zoo (Living World, Children's Zoo, Puffin & Penguin House, Reptile House, Primate House, etc.) that even on the hottest days, you can still have a good time there.
There's a brand new sculpture garden opening in downtown St. Louis on July 1. It's at Market and Chestnut and will be called "City Garden." It's supposed to have an indoor cafe, along with some outdoor fountains for kids to run around in, and of course it will have lots of sculptures, etc. Particularly if you're downtown, it sounds like it would be a nice, low-key place to stop and let your daughter play a little!