My wife and I experienced a similar trip last August, when we met some fantastic people while we explored during our short visit. We were lucky with our timing that the weather was mostly mild.
Ours was a whirlwind trip, like this one, so I would suggest that the winner should ask about extending the trip at his or her own expense to include more places to see. I don't know if they would allow extending like that, but I hope so. If I were to do it, again, I'd want more time to make sure I wouldn't miss anything I wanted to see.
If you plan to include the Smithsonian, you might want to extend the trip by weeks from what I've heard. Short of that, up to a few days' extension might be enough. For the local monuments, I'd also suggest going with an organized tour to get more background information. Then, you could go back later on your own to spend more time with particular spots you like. Or, just set out on your own in the first place to visit them at your own pace without the tour.
Even with having to pay the taxes on the prize, I would suggest that the trip could be well worth the fun and knowledge gained. My wife and I learned more in our short visit last August than either of us had learned in earlier, longer visits to D.C. with our separate families well before we knew each other.