There are so many quirky and historically significant things to see and do in London that you could spend all day, everyday in the city and still have things to see.
Some of the stranger but fun things, go to No10 Adam Street and get a selfie (it looks so much like No10 Downing St that it'll fool most people).
It's not far from Trafalgar Sq where there is the smallest Police Station in the World, big enough to almost sit one person (now disused but still there - not many know what it is even if they see it).
Climb the Monument (to the great fire of 1666).
And then there's a quick stop for afternoon tea in the cript at St Martins-in-the-field, just off Trafalgar Square (and the leaded glass window that looks like it's in a warp field).
Something that is often underated is the open-topped bus tours, might be a bit cold at Christmas but a good way to see many parts of the city and you can hop on and off as you like.
The lights across the city from Oxford Circus all the way to Carnaby St are a sight to see.
Tower of London for the history and Crown Jewels (for the ladies
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If you have a car and can travel and if it is more modern(ish) history you want;
Bovingdon Tank Museum, a couple of hours' drive to the south west (if you are a veteran you should get discount), great for the tank buff. They have a collection of Tigers from No1 to King to Jagd that are must sees (at least for me). Great place to lose yourself.
The Imperial War Museum Duxford is not far, a trip north up the M11 motorway towards Cambridge and you can't miss it (jcn 10 IIRC). There are many aircraft from the 8th Air Force on display and planes from all periods along with the land warfare exhibits. Makes a great follow-up trip to the IWM London.
That leads to Cambridge itself, another great city.
I hope you enjoy your stay in our fine city. If there's anything I can help with send me a message.