Nice post... Been everywhere and many more in several other countries, including the Brazilian one that got burnt, except for the Hermitage Museum - St Petersburg. Prefer the British Museum and the Smithsonian the most. You need several days to take a proper look at these.
The Smithsonian was built by funds from a British scientist James Smithson who never visited the USA. Only in 1904, the Smithsonian Board of Regents voted to bring Smithson’s tomb and remains to the institution he founded. They were escorted from Italy by Alexander Graham Bell, a member of the Board of Regents, and reinterred in the original Smithsonian building.
Also there's another Air & Space Museum, I forget the location but only about 45 min drive from Washington. Many things to see there including the Discovery Shuttle - you could just stand in front of it. From aircraft to space satellites , most of the things are there.