Some friends recently traveled to London and surveyed friends for ideas. This was my list.
If you're going to be spending a fair bit of time in London, get Oyster cards at the airport. That makes getting around pretty easy. And the tube from the airport is our preferred method.
You have to do tea. I suggest doing it more than once. Harrod's or Fortnum & Mason, then the other. If Prince Charles or the Queen is willing to have you over, you could go a third round. Two words for you: clotted cream.
Shakespeare was fun. Make time for whatever is playing in that teeny little theatre.
The Sky Garden requires careful planning but is worth it.
Those Brits make excellent (and a slightly different) greeting cards. I think we dropped in on a chain called Paperchase (near the Sky Garden).
A visit to Greenwich is cool. Allow a fair bit of time for the observatory. Wouldn’t hurt you to have read Dava Sobel’s Longitude.
Punting in Cambridge, but not Oxford, where they don’t do it right. The Eagle Pub in Cambridge where Watson & Crick drew DNA helixes on napkins. The Eagle and Child in Oxford, where Tolkien and Lewis drank. Who had the greater flight of fancy?
Pints and pies. Darts if you can find it. Rarer in London these days. I think they’re adopting the Irish perspective, which, according to my friend from Cork is that, “it gets in the way of the drinking.”
The British Museum! If you can stay near there, we'd suggest dropping in on your way out each day for one (or maybe two) things.
Evensong service at Westminster Abbey. They're very fussy. St Paul's is a bit more relaxed. But Westminster is better.
If you are at all interested in WWII, the Churchill War Rooms are fantastic.
If you like Indian food, London is the place for you. Dishoom is _the_ Indian place. Means "mojo" in Hindi. You'll want to get the Viceroy Old Fashioned. The black lentil dish was amazing. Nothing was bad, so go with your favorites.
The Grand Imperial London has hands down the best presentation of Peking Duck we've ever had. It's probably the traditional way, but it had never been done for me.
Out front of King's Cross station they sometimes have a market. Get a snack (or two). See if The Scottish Kitchen is there, and if it is, get the smoked clava and thank me later.