If the weather is clear, a trip in the London Eye will give you a great view of the City. For me, The Tower is a must see. The British Museum is fabulous, free, and probably way too time-consuming for your short visit. Maybe the V&A instead? A great British play would be fun, especially if Agatha Christie's "The Mousetrap" is still running. A HOHO bus ride around the most famous sites, altho pricey, might be the best way to see as many famous sites as possible in a limited amount of time.
BTW, My experience in lunching in their dining hall was also good. I tend to try out the "tourist site" restaurants in London due to convenience, time-savings when on a short trip, and have not been disappointed. Not gourmet, but usually very fresh, in my experience. I've dined at the Tower, Westminster Abbey, Hampton Court and Buckingham Palace, when it was opened to the public during a July visit.
Westminster Abbey is a favorite, and can be quite crowded at times. I highly recommend that if you go, you use the audio tour, otherwise, you won't know what you're seeing. The restaurant downstairs is quite good with fresh foods for lunch. Even in the height of tourist season, July, with elbow-to-elbow in Abbey, the restaurant wasn't crowded. No view, as it's cave-like.
The V&A is quite wonderful, but unless you want to see other places in the same area, it may be a bit out of the way for that alone, given so little time.
The Museum of London is really good.
Not too big, and gives a great overview of how London evolved. It would take you about 2-3 hours to visit.
In the evening, get theatre tickets; you can sometimes buy them at the door in the afternoon for the evening performance, providing the production is not too new or trendy.
I'm always wary of the phrase 'must see', as my must see might be your ho-hum (and vice-versa). The above suggestions are fine, but what sort of thing do you want?
For only 2 days I'd ask that you give us 2-4 "I want to feel ...." demands, and between us we can offer suggestions that might excite.
I want to be taken back in time
I want to feel awestruck
I want something that is uniquely London
The British Museum is huge but we went in once just to see the Elgin Marbles, stopping at the Rosetta Stone on the way, and it was well worth it. The display for the Elgin Marbles is fabulous, as are they.
Depending on where you're staying, you might consider the treasure room of the British Library. For a museum, it's relative small. However, you'll see the original Magna Carta, handwritten scores by Mozart, and lyrics and music by Lennon and McCartney among a variety of other things.
I'd suggest beginning your stay with one of the double decker bus tours of London. This will give you an overview of the city and a chance to "see it all," at least from the outside.
Before you go, buy a good guidebook. Make a list of all the sites/activities that most interest YOU based on your own interests (eg museums, gardens, historic sites etc). Try to set some priorities and then organize these by location so that you can make the best use of your time.
Some guidebooks are set up based on what to see if you have one, two, three, etc. number of days. You might look for one of those books to get some ideas, but the real "must sees" depend on you and what most appeals to and interests you.