(In this particular case, it was a medievalist. Tailoring the tour to a medievalist is an optional extra in this question, however.)
Where we went
A quick glimpse of the Tower of London while changing between Tower Gateway and Tower Hill
(Circle/District to Victoria. Then all walking.)
St. James Park
Houses of Parliament
Look at the London Eye from across the river
Up along Whitehall past Downing St.
A quick stop in the National Gallery to see the Wilton Diptych
British Museum with quick stops at the Rosetta Stone, Elgin Marbles, Franks casket, Lewis chessmen, and the Mildenhall Treasure
And then we were out of time.