I have a close route to yours. This is my low-traffic, low-hill route: I ride down Fairfax, left on Irving, right on 13th, left on Highland, right on Key, left on Veitch to the Custis, which runs next to Lee Highway at that point (with a detour for construction now – just cross Lee and follow Veitch to the trail).
I take either the Roosevelt Bridge or the Memorial. From the Roosevelt, I take H St. to Pa. Ave. past the White House. To take the Memorial Bridge, you go under it first and then turn right about a quarter mile past. Be very careful at that intersection if you do that, as well as at the crossing of Lynn St. (the “Intersection of Doom”).
Let me know if you want to ride sometime and get a tour of a route from a grizzled old commuter (me).