If you want to tour the White House, I think you may need to wait a long time (unless you are friends with the current occupants). After 9/11, tours of the White House ceased and they probably won't start up again as long as the USA thinks that the White House could be a terrorist target.
The resorts at Massanutten aren't close to DC, but they do offer bus tours of Washington, D.C. The Fairfield's in Williamsburg (including one that is in II) also offer Washington, DC, bus tours on their activities schedule.
The bus trips may be a good option for a first trip because you don't have to worry about where to park, how to get from here to there without getting lost, etc. You can see many more sites via a bus trip than you can on your own, but you lose the flexibility of deciding which site to see for how long. One of the trips offered at Massanutten, though, is a bus ride to the Mall via Arlington cemetery. From there you can go to anything within walking distance or you can take a trolley to popular sites; the bus picks you up at the Smithsonian at the end of the day.
Another possibility would be a direct trade with a Fairfield owner for the resort at Alexandria, VA.
Cherry blossom season (early April?) is a popular time for visiting Washington, DC. The weather tends to be hot and humid in the summer. If your children are in school, spring break might be a good choice.