"The Smithsonian" is a collection of almost twenty museums, most (but not all) clustered along a two-mile stretch called the National Mall. A full "tour" of the Smithsonian would take about a month of full-day visits. You probably have time to look at one exhibit in one museum. Is there a particular type of art or period of history you're interested in?

Library of Congress Jefferson Building tours run hourly, 10:30 a.m.-3:30 p.m., and last an hour. Tours are walk-up, first-come, first-serve. On weekends or in summer I'd suggest arriving 20 minutes early to stake your space, but mid-day on a November Tuesday you're probably fine showing up 5-10 minutes in advance. You can also just wander the building on your own if you don't want to spend a full hour or arrive out of sync with the tours.

The LoC Jefferson Building is directly behind the U.S. Capitol, so depending on when you arrive relative to the start of the LoC tour you could admire its exterior before or after. I wouldn't expect that to take more than fifteen minutes, but then I'm not a big architecture admirer and also I walk fast. Maybe give yourself half an hour if you want to walk all the way around the building and take lots of pictures. The Summer House in the northwest corner of the Capitol Grounds is probably my favorite part of the whole thing, so take a minute to admire that, too.

The White House is about a mile and a half west of the Capitol. You could take the Mall Circulator Bus (picks up at Fourth and Madison, costs $1 per person, exact change cash or SmarTrip Card only), but honestly it's not going to save you any time over walking, so I'd either walk (30 minutes) or Uber/Lyft (15 minutes). If you want to visit a museum I'd suggest one that you have to go past anyway en route, given time constraints; that includes the National Gallery of Art (not technically part of the Smithsonian), the Museum of Natural History, and the Museum of American History. Lists of current and forthcoming exhibits are available online if you want to target your visit. The National Gallery has several cafeterias/cafes, which are reasonably upscale as museum food goes, although also (as museum food tends to be) expensive.

So, an hour for touring LoC, half an hour for walking around the Capitol, half an hour for walking from the Capitol to the White House, however long you want to spend museuming en route, and half an hour to walk around the White House. Then Uber/Lyft back to wherever you stashed your luggage, and head to the airport.

Oh, and I'm not aware of any hotels in the city that offer "day rates," nor will most let you check-in before mid afternoon. (There are hotels by the airports that offer day rates, but the last thing you want to be doing is adding extra trips to/from the airports.) I would honestly go with the "metro (or Uber/Lyft) to Union Station, store bags there, walk to Capitol/LOC" option; it's expensive but low-hassle and puts you right where you want to be to start your touristing.

Final edit: If you really want to see the Lincoln Memorial and are Uber/Lyfting it to the airport, ask if your driver can take you past it on the way out of town. It should only add 10 minutes or so to the drive. (This of course presumes an Uber driver who knows their way around and isn't 100% dependent on Google Maps.)