If you want the most punch for the buck, plan to only visit London. While there, visit Windsor Castle and St. Paul's Cathedral. Take a cruise on the River Thames. Visit the London Underground and tour the quaint pubs. Walk through the residential neighborhoods for a unique experience.
I would not miss the Lake District, Hadrian's Wall, Stonehenge, the Tower of London, the British Museum, the Greenwich Observatory and Prime Meridian, Hyde Park, the Houses of Parliament, and Buckingham Palace.