Hello OP. Sorry I'm late, thanks for the ego-massage- very welcome! I'm not sure I'm the best tour guide for London, though. I stick to central and generally just work, eat, drink or shop there.
I'll do my best...
1) If you're after a 'British Experience' and don't want to do anything museum-7, then why not take afternoon tea at a posh, old-skool hotel and pretend you're a character in an Agatha Christie novel? I went for this a few years ago on my birthday, but with someone stiff and too middle class, who made it boring. Go with someone fun, and have a laugh with it- but don't be too vulgar darling, you don't want to be [italic]that[/italic] person. I went to The Goring, but I'm not sure that's the best. The Ritz one imagines would be too full of tourists. Be sure to dress up, however.
2) If you like culture, then catching a film at the BFI on the Southbank is a must. They often have seasons, and it's not too expensive in the main. Don't bother with the food in the cafe there though, it's overpriced and there isn't much of it.
3) If you're looking for somewhere different to eat at, then the National Portrait Gallery restaurant has pretty good food, and isn't too expensive either. Plus the views are quite interesting, being high up but not ridiculously so. You might spy some interesting roof terrace activity elsewhere if you've a good pair of eyes. DON'T go to the OXO Tower- I did, it's overpriced and the food and service and even the furniture are shabby. The views just aren't worth it.
4) Boozing. Soho you probably know, but if you don't I will tell you this. Comptons is exorbitantly expensive, but has a good atmosphere and is cruisy in a (generally) good way. The rest are alright, stay in the Old Compton Street area and surrounding if you want gay. If you're looking for a proper non-gay London boozer, you're out of luck. Even my favourite 'secret' ones of the last 10 years are starting to get too popular now. The ones that aren't have been gastropubbed. If you want an authentic British Pub experience you have to go out of London.
5) The Tate Modern is yes, obvious, but more importantly, free entry and it has a vast amount you can see for that; and more besides if you hand over your cash. The bar has nice views, particularly worth seeing at night, of course.
In all honesty, if you're after something a bit off the beaten track, ask someone else who actually lives and breathes the place would be my advice. But thanks for thinking me your go-to person, even if probably incorrectly so.