In DC, the Mall can be fun for a little one. The monuments, which have great sculpture, and which little ones can sometimes go up and touch are great (the map of the Navy Memorial, and the soldier which you can go up to, is great - but be careful of the fountains and such).
The National Gallery of Art has a sculpture GArden with a fountain, lots of benches, and a restaurant. Many of the sculptures are whimsical.
Several of the museums have interactive stuff that might be of interest to little ones. The National Museum of Natural History is one, the caveat being that the stuffed animals, dinosaurs and such might be a little intense for such a little one.
Gadsby's tavern in Old Town Alexandria has a regular Toddler and Me session where they play with the toddlers in authentic 18th century ways while parents get a chance to enjoy a cup of tea or something.
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