I hope you are saying that you already have your plans for the White House, either as part of a group, or as a guest of your Senator/Congressperson - since those are the only ways to see the White House right now.
You might visit the Capitol Building or the Supreme Court. Tours of the Capitol begin at 9 AM, and the Supreme Court at the same time, but only Monday through Friday.
Walk the Mall - the memorial walk from the Washington Monument past the World War II memorial, Vietnam Veterans Memorial, Korean Veterans Memorial and Lincoln Memorial is a great walk in the evening - when these are lit up and when other things are closed.
Pick a subject that interests you and visit a Smithsonian Museum or two. While the Natural History (Dinosaurs, Gemstones, etc.) and the Air and Space Museums are top draws, you might also visit a less visited museum. The American Indian Museum is fantastic, and art museums including the Sackler and Freer Galleries (Asian ARt) the HIrschorn (Modern Art), American Art and Portrait Gallery, Renwick Gallery of American Crafts and the African Art Museums are worth considering. The American History Museum and the ARts and Industries Building are under renovation.
You might visit Ford's Theatre, where LIncoln was assassinated, the Kennedy Center (which has a free concert every day at 6 PM), the Postal Museum, the Botanical Garden, the Arboretum. Oustanding private museums include the Spy Museum, the Holocaust Memorial Museum, and the Newseum, opening this summer, as well as private art museums like the Corcoran.
Get out into a neighborhood. Real washington DC is very much a city of neighborhoods. Think about walking around Chinatown, Shaw and U Street, Adams Morgan, or if you must the more touristy areas of GEorgetown and Old Town Alexandria.
Visit Arlington National Cemetary, and see what the price of freedom is about, and visit the Kennedy Gravesite and the tomb of the unknown soldier. Go to a baseball game at RFK stadium, or check out the great dining and theatre of the Penn Quarter. There is much more.