Honestly, you really don't need a hotel "close to stuff to do." The underground in London is cheap, reliable, and pretty fast. I'm actually headed to London and Wales in March (and I've been a few times already, family over there), and I plan on using the underground for almost everything. Driving in that city is just asking for trouble.
As for things to see.. do you want to do the normal tourist-y stuff? If so, definitely see London Bridge and Tower Bridge (a cruisetour on the river is pretty cheap), Buckingham Palace (if you're into that, it was pretty neat I guess), the Tower of London, London Eye, Big Ben, Westminster Abbey, etc. etc. I really like Leicester square, personally, for some shopping and people watching, but the most fun for me is just to take the underground to a random stop, get off, and see what I can find to do. The open-air double decker bus tours are good to see a lot of the historical/famous bits of the city, but I'm not sure how fun that would be in February.
Have a great time! If you want any more suggestions, just PM me. I've been looking into this stuff for a few months now for my trip.
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"Without passion man is a mere latent force and possibility, like the flint which awaits the shock of the iron before it can give forth its spark."
— Henri-Frédéric Amiel
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