Train to Paris is very straightforward from central London. Not cheap, but you save a bit if you buy significantly early. It would also make for a long day, but as long as you’re fine with that it’s doable.
Museums in London are the best economized activity, as they’re free! My favorites, all centrally located, are the National Gallery, British Museum, Victoria & Albert, Tate Britain, Courtauld Gallery, Royal Academy, and Wallace Collection. If you have less conservative tastes you can try the Tate Modern and Serpentine Gallery.
Plenty of “tourist” activities in London are also fun. I particularly enjoy the Tower of London, and Westminster Cathedral.
The Ledbury is meant to be the place to dine, along with the Chiltern Firehouse.
For wine you might visit Hedonism, which is widely over-priced, but has arguable the most opulent visible store inventory of any wine store in the world. Their enomatic machines also give you an opportunity to try all kinds of rare and expensive wines, for much less. The Sampler also has a great selection in their enomatic machines, and somewhat better prices. For history, visit Berry Brothers & Rudd. Still family owned and run, with a store on St. James’ Street – one of the toniest addresses in London.
Beyond that I think activities will depend on your interests. I love gardens and parks, so I always enjoy King James Park and Hyde Park. There is also plenty of world class classical music, ballet, and opera. London is also a great city for simply walking. Especially if you begin somewhere like Bloomsbury, and walk towards the river.
Finally, even in London you can find superb examples of classic British stately homes. If that’s something that interests you, let me know, and I’ll recommend a few.
*all written from the perspective of an American.