I'll be honest; despite living 30 mins from London for most of my life I'm utterly useless at naming anything good to do there. Also February in London is usually pretty cold, can be wet and even snow if you're (un)lucky. But then it has snowed in May in recent years so it is a complete lottery.
I agree with Barry though - Camden Town is a cool place to go no matter what time of year it is, and it is pretty accessible on the London Underground too. If you're planning on using the London Underground a lot, you can pick up an Oyster card at most tube stations - it's basically a pass you top up with money and you just check in and out every time you use public transport, and it usually works out cheaper than buying tickets on the day too. It's worth Googling it if you want more info.
Generally London is pretty expensive, even by UK standards. Drinks are usually at least Ł1-2 more expensive, and meals are Ł5-15 more expensive than outside the capital. Most places usually do deals though, especially early-bird or Sunday to Thursday. Plenty of drinks places do happy hours as well. A good rule is that anywhere right next to train stations, tourist attractions and theatres will charge premiums, and the further out of the centre you go, the cheaper things tend to get - this is pretty common sense but lots of people don't realise this. There isn't that big a tipping culture in the UK - not many places will charge a mandatory service charge for groups of 6 or less, and no one will be particularly offended if you don't tip, but 10% is the absolute maximum if you've had decent table service. Most pubs expect you to order your food and drinks at the bar and pay there and then.
If you have laptops or smartphones or tablets, most places in the capital have free wi-fi - McDonalds restaurants and Wetherspoons pubs are safe bets if you need wi-fi in a bind, but loads of pubs, cafes and restaurants use 'Cloud wi-fi' - you normally just need to create a free account and you can log in wherever you go.
London has loads of parks and gardens. I couldn't really tell you which are the best ones but I'm pretty sure Google will be your friend again.
What I would do is a few weeks or days prior to arrival, Google for things going on in London on the week of your stay - there's always something going on, a lot of it will be free or good value and you'll be bound to find something you enjoy. If you want to go see a musical or a show, there are stands around the West End that can sell discounted last-minute tickets, sometimes at a 90% discount. If the Book of Mormon is still running and you haven't seen it, I can honestly say it's one of the best shows London's ever put on.
I am in the UK during those dates but due to work commitments will probably only be able to make either the 18th or 19th if you wanted to meet a fellow TPMer. Not sure about Becki, Kirsten or any of the other TPMers.