Due to its proximity to the Union Station Metro station, the Hotel George is, IMHO, very conveniently located. You are two stops away from the Gallery Place/Chinatown area which is teeming with restaurants and shops plus the new Spy Museum (very popular; get tickets in advance).

One more stop away is Metro Center which is probably as close as you can get to the rear of the White House which is the most photographed side of the White House.

Two more stops away and you are in Dupont Circle which is a very popular neighborhood with lots of bars, clubs, shops and restaurants. There are a lot of hotels in Dupont Circle too, which is a good alternative to staying downtown.

Metro also takes you to the Smithsonian station which is ground zero for all the museums, the Washington Monument and the new WWII Memorial. Note that all the other memorials (Jefferson, Lincoln, FDR, Vietnam War, Korean War, etc.) are a long walk from any Metro station; Smithsonian being the closest.

Locally, the area around the Hotel George is not too lively but pretty safe. It's a short walk away from the shops/restaurants at Union Station and Capitol Hill.

You may also want to look into the Hotel Monaco, sister hotel to the Hotel George. It is centrally located in the Gallery Pl/Chinatown area which is a very lively and busy area even at night. Because Metro takes you everywhere you need to go, I think picking a lively area is more important than staying as close as possible to the sights which are in areas that have very little activity at night.