London is not one atmosphere. Westminster is ceremonial, the City is a financial grid that empties at the weekend, Southwark feeds you, and the residential boroughs are where the city actually lives. Treat it as a cluster of villages with unusually good museums between them.
General admission to the national museums is free. The paid sights are optional, not a syllabus. Stay in one area, walk farther than you planned, and use the river as a compass rather than a to-do list.
See
National Gallery
One wing, not the building. Early Italian or the Impressionists, then leave through Trafalgar Square before you are tired of both.
Tate Modern or Tate Britain
Modern if you want the building and the Thames; Britain if you want Turner without the turbine hall.
A market lunch
Borough for the famous version, or somewhere like Broadway Market or Ridley Road if you want London eating for itself.
A park with a view
Primrose Hill, Greenwich Park, or Hampstead Heath. All three outclass most paid viewing platforms.