Lived in London all my life. Only just moved away after 50 odd years. New things to do still seem to pop up weekly.
I would stick to one area to explore rather than try and travel all over if you're only effectively there for a day once you've factored in travel and dinner/ show/ drinks etc.
So London Bridge: The Shard, The old Operating Theatre, Vinegar Yard, Bermondsey street, The Bermondsey beer mile, Borough Market, The Clink, The Golden Hinde, HMS Belfast, Tower Bridge, Tower of London. A ride on a Thames clipper. Not suggesting you do all of them but it's all within 1/2 a mile of each other.
Spitalfields: The market, Hawksmoor's Christchurch, the silk weavers' houses around Fournier St. Any amount of Jack the Ripper walks, Brick Lane for shops and a curry, great bars, Dennis Severs' house, Bunhill Fields if you like cemeteries, it's not far from Liverpool Street, The Victorian Bath House, Sushi Samba etc
Marylebone: The Wallace Collection, Daunt books, Marylebone High St, The Planetarium, Tussauds, Labour and Wait, short hop up to Regents Park to go boating on the lake...
Bloomsbury: The British Museum, Corams Fields, The Foundling museum, St George's Church, The Brunswick centre, The cartoon museum, The Charles Dickens museum, dinner at Circo Popolare or Dalloway Terrace...
Obviously there are the big draws like the HoP, Westminster abbey, Buckingham Palace, The big museums and galleries. All of which are wonderful and some need best part of a day to appreciate. But any single part of London has tens of little museums, galleries, oddities, churches you can just wander into, interesting bars, coffee shops that aren't chains which are run by people who love coffee, independent shops, food from every part of the globe...
I miss it. I'd love to be able to see it through the eyes of someone who hasn't been in 20 years!