Spread across 4 floors and a glorious atrium, the British Library is seriously the best place in London to work productively. And it's certainly no secret. I frequent there for long sessions of laptop work and have counted over 1000 people - students, freelancers, academics - setting up work stations there every day. The vibe is enterprising and motivating, to say the least. There's even a nice cafe and restaurant and an exhibition hall if you fancy having a break.