No trip to London is complete without a trip to Dr. Samuel Johnson's House in Gough Square. It's impossible to leave without a strong sense for the man and for his life in London, and the house is a beautiful sight to behold in its own right (and can be a great place to escape the hustle and bustle of Fleet Street for a few minutes).
If you're hungry (or thirsty), you can visit Dr. Johnson's favorite pub, just around the corner—Ye Olde Cheshire Cheese has a portrait of Samuel Johnson hanging in the dining room, just above his favorite seat (a pilgrimage site for many authors and writers, including Charles Dickens). Make sure to note the sign at the door explaining that the pub was rebuilt in 1667, following the Great Fire of London.
On the way, you can pay tribute to Hodge (Dr. Johnson's cat), who is immortalized as a statue in Gough Square with "a very fine cat indeed," inscribed upon its base.