London Eye: Huge observation wheel giving passengers a privileged bird's-eye view of the city's landmarks.
Buckingham Palace: Home of the British Queen & State Rooms
British Museum: Huge showcase for global antiquities, including Egyptian mummies and ancient Greek sculptures.
Hyde Park: Huge green space, home to Diana Memorial Fountain, with boating and swimming in the Serpentine lake.
Trafalgar Square: Nelson's Column, fountains & lions
St Paul's Cathedral: Iconic cathedral's churchyard & garden
Shakespeare's Globe: Oak-and-thatch replica of original Elizabethan theatre, showing Shakespeare plays in the open air.
Churchill War Rooms: Secret underground headquarters where Prime Minister Winston Churchill lived and worked during WWII.
I especially recommend Shakespeare's Globe, where Othello, Hamlet, As You Like It and The Winter's Tale are scheduled to be performed in the summer theatre season.