Ahhh... I'm a Londoner born and bred, well until I left, which wasn't
that long ago.
I would advise finding a cheap hotel in Central London. London is a walking city, and you'll benefit from being able to explore and then crash. Not sure what the rates you have at LGW are like, but there are some cheaper places in London, some netsurfing will identify them. Sidestep, Expedia etc...
If you're interested in some of the History, there's a group called
London Walks that runs some very interesting tours. The history of London is fascinating. I always said I would only leave when I stopped learning about London. Unfortunately practicalities got in the way of that ideal, but I still enjoy going back to continue the learning.
As for an evening out. Food is generally more expensive than the states. Beer does not have to be, if you find the right pubs, basically those slightly off the beaten path without enormous queues of tourists outside. If you are going to be staying in Central London, drop me a note and let me know where, and I'll give you some pointers on the area, if I know it.
The liveliness of the area around Covent Garden, Leicester Square, Picadilly Circus are fun to take in the atmosphere, night life or just an evening stroll, as is the London skyline from any of the bridges (or the London Eye which is well woth the money, day or night!). There is now a riverside walk, which runs mainly along the South Bank, has a number of fantastic pubs along it, quite a few of them with a Shakesperean connection, and a conveniently located pizza restaurant not far west of the South Bank Center with a super view.
If you're going around Christmas, the Regent Street lights are themed along with all the store fronts, which can be a smile for an evening stroll.
As always I'd recommend the Tower of London as on of the best tourist spots. It's been around for a major part of London's history, and a tour around it reflects that.
Alright, I'll stop waffling on now... if you want any more information, feel free to PM me.