On a hot day like today, take a wander down to the National Mall, and I guarantee you'll see some prime tantrums being thrown. "It's too hot!" "My feet hurt" "I want ice cream" "I don't care what's in that building, I'm sick of museums!"
And this is just the parents.
So, before your kids turn into a molten pile of goo and your wife gives you that look, take some time to relax, buy them (and you) some ice cream, and maybe even splash around a bit. Here are some options:
The American History Museum, Constitution Avenue Side - It's not shady, and it's kinda exposed, but the fountain out front was originally designed to allow people to splash around a bit. A some point, someone half-heartedly plunked some fences in there to keep people away from the geyser-like fountain, but you can still dip your feet in and cool off. Ice cream is readily available from curb-side vendors, but they have to leave for rush hour at 4 pm. This is a good choice if you plan to walk on over to the White House or downtown afterwards.
Perhaps a more restful option, a little better protected from busy Constitution Ave, is the Sculpture Garden. Officially part of the National Gallery of Art, it lies between their West Building and the Natural History Museum. Designed as a place to showcase large and whimsical pieces of art, it's a restful place with ample benches and shade for folks to get off their feet for a bit. Even a passing school group or two can be absorbed without disturbing your rest (too much, that is). The Pavilion Cafe is actually not a bad place to grab a bite to eat, although be prepared to shell out roughly what it's going to cost you to send your kid to college for a semester. If you want to sacrifice a bit of quality, the ubiquitous ice cream vendors can be found across the street by the Archives. While outright swimming is not allowed, the guards don't normally care if you dip your feet in the water for a bit. And yes, it is a requirement to get a picture of you appearing to relieve yourself. Don't feel awkward. Or particularly clever.
If your plans include walking over to Capitol Hill, you might want to swing by the U.S. Botanic Gardens. Their National Garden is a great little enclosed space that quite often you will have all to yourself. Sadly, the ice cream trucks are no match for the Capitol Police, and your ice cream options are limited.
I'm almost loath to tell you my last cool down spot on the Mall (you're not getting my indoor rest spots!). Try to be worthy of this one. Behind the Smithsonian Castle, and technically the roof of the African Art Museum and the Sackler Gallery of Asian Art, is the Enid A. Haupt Garden (not to be confused with the Haupt Conservatory in New York, Enid got around). The horticulture is stunning here, even for a philistine such as myself. The Moongate garden towards the west echoes the Freer and Sackler Asian galleries towards that side, and the east side tends towards a Moorish theme reminiscent of the great gardens of the Alhambra. The fountain pictured is on that side, and is the very one my oldest daughter played in until her diaper swelled up like a baboon. Pictures of that will be reserved for blackmail purposes when she's a teenager.
Ice cream may be available from vendors on Independence Avenue through the Renwick Gates, based on a never-built drawing of architect James Renwick and completed a century and a third later by Constintine Seferlis. Or you can buy ice cream in the Castle and smuggle it out. It's what I do.