From my personal experience of studying in London and attending the guest lectures that go on at the LSE, I would highly recommend going for LSE.
I attended a three day mini-lecture series of Nicholas Stern, former World Bank chief economist and former second permanent secretary of the Treasury, and his outlook on environmental economics and the 5 years since the Stern Review on the Economics of Climate Change. When the Occupy movement was outside St. Pauls Cathedral, Harvard professor Michael Sandel gave a talk on 'morality in the markets'. So in addition to so much going on at the school of economics, if you are willing to find out about life then you can't go wrong in London. If you enjoy art then across the road is the Courtauld Institute and gallery at Somerset House, all the think tanks in Westminster - Adam Smith Institute, Policy Exchange, Institute of Economic Affairs etc.
Outside of studying it is a much better life to lead. Just head down to Imperial College and check out the Science Museum or pop along to the King's road off Sloane square and check out some of the most beautiful women you will ever see.
The LSE has a much better international reputation and you are likely to meet a lot more people whom are from many different backgrounds and if you go for a drink at the LSE pub, the George IV, you might end up talking to someone who studied at Harvard and they'll let you know what it is like studying in the U.S. for example.
Of course, London is the best place for internship opportunities; Civil Service, the City, and so much more.
The LSE will give you a great platform for networking.
I think LSE will give you a more broader outlook to be more pragmatic with your history degree.
I have always wondered if I should have done a history degree and commentate like Niall Ferguson.