There's a fair bit here that other people have mentioned but never mind!

The tube generally stops between midnight and half past so watch out if you get a hotel a fair way away even if it is on the same line. There are plenty of night buses which are fine to use as long as you know where you're going.

An excellent app I use is CityMapper (works for NYC too but not quite as well) which will give you options of how to get places with which tube or bus stops to use, timings and costs as well. It's so useful. The buses generally now all have displays to tell you which stop you're approaching which is nice. Try and get on a new fancy Routemaster bus if you can

St. Paul's is definitely worth a look and you can go just inside to have a quick peek without paying, or if there's a service on you can go further in and sit down. The singing there sounds incredible.

The Globe is absolutely worth a visit if you can - the outdoor theatre isn't open over the winter but I think the new indoor theatre (lit by candlelight) has just opened and started showing if you can manage to get tickets and it's your sort of thing.

It's a quite nice walk along the Thames by the Southbank and if the tide's out there are bits of somewhatgrimy beach that can be investigated

Brick Lane is entertaining, and has decent bagels, and there are often fairs and events going on in the Truman Brewery. Very close to this is Spitalfields market which can have some interesting stuff, and Columbia Road flower market which happens on Sunday mornings and gets absolutely packed at about 1 when the stalls start reducing their prices. Really nice atmosphere and there are plenty of shops around it that are fun to poke around.

Carnaby St is quite nice like Ben says, and just at the top of it is Liberty - a really old department store in an amazing building. I don't tend to buy much at all there but it's gorgeous to look around, and does a pretty good afternoon tea!

The Barbican is amazing (I may be slightly biased as I live there but it's really cool) just as an estate to wander around, but it also has stalls and things inside the arts centre, and has various stages and whatnot for shows.

A couple of places I've been to recently that are ace for cocktails and food:
The Nightjar
The Clove Club

Can't think of any more right now, but ask if you've got any questions or think of stuff you want more specific recommendations for!

~Zeeb