All the usual tourist spots: London Eye,Covent Garden, Tower of London, Buckingham Palace, Westminster Abbey, St Pauls Cathedral, Oxford Street, Carnaby Street, Harrods, London Zoo, Selfridges, Green Park, Hyde Park, Regents Park etc etc etc
Then I would add: a trip to the theatre, sit in at a trial in the Old Bailey, go to a football match (plenty of teams to choose from), Hampstead Heath and the pubs Jack Straws Castle and The Bull and Bush (from the music hall song), go to as many museums and art galleries as you can, visit all the main line stations, go to as many Monopoly board properties as you can, visit the original Scotland Yard..........
In general I would say - think what people will ask you if you've done or been somewhere and you'll be able to yes, if you'll be able to say no and regret it then it should have been on your list! Good luck!