Get the Rick Steves book. It will save you a lot of trouble.
Do NOT skip the Museum, the Abbey, the Tower, St Paul’s, or the National Gallery. I stupidly missed Tate Britain and was underwhelmed by the Tate Modern. I don’t know if that was me, poor presentation, or the late hour I went. I’d have done the Parliament tour, but I went right after the election and it wasn’t running. You have to call ahead for that. I looked at Buckingham Palace but skipped the changing ceremony.
You might consider the British Library as well. It’s hard to get to now, because the nearest Tube station was bombed, but it has a good collection of rare holy books, the Magna Carta, original manuscripts, and documents related to a host of literary and scientific giants.
I enjoyed Greenwich. The Cutty Sark is dry-docked there and worth a look. The Nat’l Maritime Museum was a little too touristy in my opinion, and it did not play well to my fear of heights. But the Royal Observatory, a Wren work, was good and the park was beautiful. And I had a bacon butty there. It’s not much but I enjoyed it. Day trips I didn’t take, but will someday, are Hampton Court Palace and Kew Gardens.
Instead of the expensive dinner, why not an expensive tea? You could go to the Ritz or Fortnum and Mason’s. Get the frou-frou moment you want, in a heavily British way, and still have time to dawdle around the theater at night.
A few personal experiences:
Don’t bother with TV. All I found was American imports, repetitive news, and the obnoxious “crazy frog” ad that’s driving Britons insane.
Most places have a cafe that smells like A-1 steak sauce. Poet’s Corner in the Abbey is particularly strong. If you like things smothered in brown sauce (like me), well, it’s all gravy.
Get the Queen’s ale at the Tower. Very strong and very smooth, and in a huge bottle that will leave you tipsy for your trek through the White Tower.
There are Indian restaurants everywhere, and they are scrumptious. I ate a wide variety of ethnic besides native British and Indian - Japanese, Indonesian, Moroccan, Greek, Belgian - and never had a bad meal. Everything is ordered a la carte, though, so the prices can stack up. If there’s a deal where you can get X number of courses for a cheap price, consider doing that.
The Museum let’s you take pictures if you don’t use flash. I got some great shots, and would have had more if I’d had the right speed of film.
There’s a Rodin sculpture immediately south of Parliament that a lot of people miss.
Harrod’s is only worth the food court. The rest of it looks a lot like a Dillard’s. Harvey Nichols is better, but still not worth building a trip around.