British Museum is about a 20 minute walk, or two stops on the Picadilly line to Holborn, or you can take bus 73, 390 or 10.
Big Ben, National Gallery, London Eye and Buckingham Palace are all around the same place. 40-50 minute walk from the hotel, or a few stops on Picadilly (Leicester Square) or Northern Line (Charing Cross branch to Charing Cross), of the 10 bus.
Tower of London is out east. Circle line around to Tower Hill.
The
journey planner from TFL is handy.