Make sure you’re on the right train/in the right part of town, though – you can see Battersea Power Station as you travel in or out of Waterloo Station.
Of course they’re safely kept in London! Everyone else who had them are foreignors 
If your partner likes naval things, take a train down to Portsmouth for the day; there are direct trains to Portsmouth Harbour.
[On the discussion of theatre: I prefer the West End to Broadway, although stunt casting on both sides of the Atlantic can be annoying.]
Another little cheap and interesting diversion is the monument to the great fire of london (Pudding lane, of course).
It is about 10 minutes walk from St. Pauls on the way to the Tower of London (another 5 minutes further on) and for an adult it costs about £4 to walk to the top. You can get a combined ticket for £10 that gets you into the Tower bridge as well (you get to walk along the viewing area along the top of it.
This. Anytime I would travel, I would pick a point a few miles from my hotel and just walk. If you’ve never been before, you’ll see cool things wherever you go in London. Last time I was by Kings Cross and would take different routes to the Thames, trying to have a pint in every pub I passed.
Thirding this advice. And if it’s too far to walk, take the bus! We hardly ever take the Tube when we’re in London now…after all, it’s full of tourists! 
Taking the bus is a great way to get a sense of the geography and architecture, without overtaxing your legs - a trip that is just a Tube station away can actually be quite far.
Still haven’t seen the river then?
OB
I’m often surprised at how good London pubs can be without them ever being anything particularly out of the ordinary. One that sticks in mind is The Blue Posts - between Piccadilly Circus and Leicester Square (i.e. right smack in the middle of one of the most touristy bits of London) - and yet it’s just an ordinary, honest, cosy little pub with wooden board floors, mismatched chairs and china dogs on the windowsills. I can’t quite work out how it hasn’t been destroyed by the modern world.
OK, so I’ve carefully examined all your suggestions and… well, seems we’ll have to move there for the full tour. 
Seriously, if it’s as fantastic as some of you are making it sound, this probably won’t be our last trip.
So now with all these wonderful recommendations for what to see, what should we avoid? Lars Aruns already mentioned Madame Tussaud and the London Dungeon. Wasn’t planning on going to see wax copies of people, anyway, but is there anything you, good people of the SDMB, have found disappointing (in London, that is)?
If you are a WW II buff, visit the Churchill War Rooms - the underground bunker Winnie used during the war. Quite impressive.
I enjoyed Greenwich, and I’m not a Navy buff. Definitely the British Museum. The Rosetta Stone is there as well as the Elgin Marbles. I’ve been to the Acropolis, and I think they’re safer in England.
Definitely agree on going to the theater. Saw Cats there before it came to NY, and took the kids to the Wind and the Willows. I’m a New Yorker by birth, and have been to Broadway a ton, and London is just as good.
One more thing - visit Hamley’s Toy Store.. Most impressive toy store I’ve ever been to, which includes the NY F.A.O. Schwarz. The toy store in Hogfather where Death plays Santa reminded me of Hamley’s - and I’m sure it was based on it.
Love this! So first thing you will want to see are key attractions that make up the rich history the UK has on offer, so typically this is Windsor Castle (great goofy pic here in front of the castle!), Stonehenge (one of the worlds most ancient historical sites) and the stunning city of Bath (pretend to take a dip in the ancient Roman Baths!).
https://www.comparestonehengetours.com/product/Windsor-Stonehenge-and-Bath.html
Then there are some quirky places to go - Madame Tussauds, where you can pretend to kiss Prince William in front of Kate!!! Or you could take a trip on the London Eye and encourage everyone to do a mannequin challenge!
Given that the OP said “The End of October” I’m pretty sure she’s there and back by now
October of 2014, no less. Thread closed.