It's rather beautiful in its mesh of old and new, and being huge and sprawling means that tourist are not crowded so densely in a few areas, but spread out over a few miles. Yes, there are tourist stopping erratically to take selfies every few steps when you hit a big monument, and people with cameras held up on sticks narrating videos while holding up the working masses on Oxford Street (whom I have so far successfully suppressed the urge to punch) and a few drunk party bikes holding up all manner of traffic, but for the most part, you can be in London and feel like people live there and work there, and with a little bit of an impatient pace, you can feel like you are part of that rather than just an observer.
Besides, winter in London isn't nearly as cold and blustery on the east coast. It's damp, and frequently grey and featureless, but it's only been icy a few times in the mornings so far, and nary a snowflake to be seen in years in the Southern part of the country. It's good advice to dress warmly and expect rain, but it's no where near as terrible as it's made to seem, as the locals complain noisily at each drop of a degree on the thermometer. But ah, green grass! In January! It makes things so much less bleak.
A couple things were new to me: on the way from the Tube to the Tower of London, I happened to glance down and alleyway and spied something interesting, which ended up being a bit of the original Roman wall that surrounded the square mile of the city, now surrounded by hotels and large-scale construction projects, but somehow tucked away and preserved and infrequently noticed.
It's a sprawling complex inside the walls, with lots of towers and odd rooms to explore, and a really hyped exhibit of the Crown Jewels where they plop you onto a moving walkway to spy, keeping the lines industriously moving.
For over 900 years, the white tower has been a famous landmark, revered and dreaded and the site of unjustly imprisonments and the executions of queens.
Today, it's a working museum, with loads of military types trotting around with much ceremony.
The museums are the best attractions in winter, and people are always a bit in awe that they are all 100% free (well, except for exhibits, and those cost a mint) and you can just walk in off the street into the enormous turbine room at the Tate Modern. Even if you don't really fancy modern art, it's still a site to see: a repurposed power plant, gutted outfitted with galleries and enormous bespoke exhibit spaces.
I've found that a week here will let you scrape the surface a bit: see the main sites, take in some museums and culture, learn to love (or hate) the local beers, sample a good curry and seek out some decent pub grub. If you have the luxury of spending two weeks in London (you enviable thing!) you can spend more time exploring the sprawling parks and hidden crags of nature if the weather permits, and start to get a feel for how each neighborhood feels like an individual village that just happened to get swallowed into a giant metropolis.