I picked Sciences Po because it is allows studying lots of different stuff (politics, history, economics, sociology, law, languages etc.)
Also because I want to study in France, the uni is very international, lots of cool societies and associations on the campuses etc.
Assuming you're British, the price is 9250 at LSE as opposed to 12200 at Sciences Po, which is more expensive. As you say though, London is expensive, and keep in mind you might have to commute every day unless you get an apartment in central London, which would be very pricey.
I'm going to Poitiers and I've got a great apartment just across the street from Sciences Po for a reasonable rent so I personally prefer Sciences Po.
You should also consider your future; both unis are very well known in the realm of politics and the social sciences, but if you're planning on living and working in the UK, LSE is more familiar to most. Sciences Po, meanwhile, is very well known in France. Obviously you don't know exactly what you'll be doing and where, but it's worth considering.
Also a major thing for me is the year abroad - my third year at Sciences Po will be abroad, and is integrated into the course, whereas if you want to study abroad at LSE you probably have to add a year.
You've got time, just do research on both unis, maybe make a list, talk to friends/family/uni counsellor, and figure out which one appeals to you more.