The Smithsonians are all free. Many others are free as well. You may or may not need timed tickets to enter some of them, or for particular exhibits (just a crowd control measure). Definitely too early to try and get them now.

Lincoln Mem is good (you can walk right in); Wash Mon needs timed tickets, I believe (it’s a very small elevator to a very small space). I don’t know if they still do walk down tours, but those are very cool. All the memorial stones are inside the obelisk.

Go to the Old Post Office Pavilion (the Waldorf Astoria) on Penna Ave and go up their clock tower; best view in town.

As for getting a WH tour, check with the embassy, they may be able to set something up.