Hi, looking for expert and / or local knowledge.

Family of 4 visiting DC tourist sites next week. We have a 1 AM arrival IAD, so I booked a room at Dulles Country Inn, but was recently notified of 11 AM White House tour that same morning. We're booked for the Willard later that day but not when our flight arrives as they're sold out. We can wake up, go straight there and drop our bags at least.

It looks like the main choices to get there would be airport taxi, uber, and metro. What would be your choice and what time would you suggest leaving the Country Inn that morning. I'm assuming rush hour goes to about 9 am? Would leaving then be cutting it to close? Is the metro, with all the construction and bus connections, going to be any better than sitting in rush hour traffic?

Your advice is much appreciated, thanks!