To the east is the Capitol Building, which looks spectacular at night with the dome lit up. In the center is the Washington Monument, towering over everything in DC. It is on a hill, but doesn't photograph well at night.
From the Washington Monument, walk south around the "Tidal Pool" to the Jefferson Memorial, and then continue in a clockwise loop, you will reach the FDR Memorial, (future site of the MLK Memorial), the Korean War Memorial, Lincoln Memorial, Vietnam Memorial (better visited during sunlight), WW2 Memorial (nicer in spring/summer/fall than in winter), and back to the Washington Monument. There are a few other small statues here and there. From the Washington Monument, the White House is to the North. The southern view is blocked off pretty well due to security but you can get a closer view from the north side.
Though areas can get dark and night, there are a lot of cops around and I never felt unsafe. A few areas such as the FDR Memorial are a little isolated. Public restroom in the Lincoln Memorial.
WARNING! Hiking and outdoor related sports can be dangerous. Be responsible and prepare for the trip. Study the area you are entering and plan accordingly. Dress for the current and unexpected weather changes. Take plenty of water. Never go alone. Make an itinerary with your plan(s), route(s), destination(s) and expected return time. Give your itinerary to trusted family and/or friends.