We always park at the Reagan Trade Center whenever we visit the National Mall. It's centrally located, has ample spaces, is not too expensive, and is open 24 hours. It's a 15 minute walk to DAR Constitution Hall and you'll get a nice short walk through DC.
If you're willing to gamble, you can try the street. But, if you don't want to risk 30 minutes of one ways, a garage. I used spot hero last time I went, and it was straight forward, but kind of a hike. All the nearby garages are daytime weekday focused.
Garage.
1700 block of D St. NW
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