I should probably preface this by saying– The times I've been in London I didn't have any guidebooks or a map or a GPS or read anything about England before I went. Thanks to my American public education I pretty much knew that there was King Aurthur, the Magna Carta where Prince John was beheaded by Jack the Ripper and Winston Churchill was put on the throne; then the USA saved their asses in WWII and then we all had a laugh at the French for a bit. I pretty much just took the tube to a stop that sounded like a place you'd want to go, hopped off and went into the closest bar, ordered a drink and asked what was about to look at.
I haven't been to London since the changes to the area but I always like taking the tube to the London Bridge stop and walking about the southeast end galleries and bars. When you get out you can duck back under some tracks and overpasses and you step back in time. Back in time to a time when they paved over the cobblestone streets that one time, maybe in the 1950s and now all the cobblestone street are showing through again. There are churches next to butcher shops, where I think they actually kill the animals in the alley out back. There are pubs open pretty much all the time.
There's the Golden Hinde, old boat replica and St. Paul's which is pretty cool along the Thames near the LB Station. Better- there's a great pub with a small outdoor deck right on the river right by it if you get a nice day and want to throw back pints. All the stuff in that neighborhood, from the butcher shops to whatever else goes on under the overpasses and rail tracks is pretty interesting and tourist free. Old churches and great pubs. Feels like old England, or what I imagined England to be like in the olden days, except slightly cleaner.
If you head back west up the Thames, the Tate Modern has a collection not to miss if you're in to modern art. I've only been to the old Tate, before they got their new spot, but it should be even better now with a proper sized gallery. Otherwise, there's a Shakespeare theater thing and the Clink(?) prison museum around there that wasn't too terrible.
If you head east from London Bridge station, past the Tower Bridge, Winston Churchill's (experience WWII museum??) looked too cheesy for me to go in, but I heard it wasn't bad either. If you or your girl is into modern design and fashion- there's the Design Museum east of Tower Bridge which is different and pretty cool. I thought the science fiction museum was around there too (? forget where exactly)... lots of Daleks. Brits and their Dr. Who... just south of the brit rail station between London and tower bridges is the fashion museum, which isn't worth seeing unless you have a design major along with you, but Delfina is good for lunch... if you don't mind spending 30brit pounds a person for lunch.
I walked most the rest of the east end and there's some cool little neighborhoods, good pubs and the like. When I was there, there were also signs from the Brit police that said "don't walk about in this neighborhood; we don't" but other than a few skinheads and shifty looking kids I didn't run into any problems. Maybe not some place to take a girl, and maybe they've cleaned up that area now... been a while. I've also found, with a shaved head, DJ or courier bag, sh*t jeans and a pair of trainers-- everyone else thinks you're shifty looking kid and assumes you belong there. Works from London to St. Petersburg. I mumbled a bit and talked fast and most people didn't even notice I was American.
Other than that, if you want to skip the tourist traps... I hate to say it... Notting Hill was a nice time. Open air markets... pubs with live DJs at noon. Bars tended by cute and approachable girls, rather than burly, smoking guys.
Couple other items:
I made it out to Sherlock Holmes's house then realized it was the imaginary address of an imaginary person and didn't go in.
The National Gallery also had zero line when I was there in the height of tourist season and it had an awesome collection from (Renior I think?) a major artist on traveling display.
I missed most the swords, Jack the Ripper, old timey history stuff the times I've been, but I usually drink through lunch and wander through museums until somehow I end up arguing with a Kiwi about rugby or something else I know nothing about in a dive bar that doesn't appear on maps or have a sign outside. I saw a really great violin concert along the Thames in a covered alley way for 3 pounds and half a pint of whiskey. Come to think of it, it wasn't a concert as much as it was a group of Romanian gypsies. They were nice... didn't rob me. Much better than the X dealers I played pool with. Or was that snooker?