From what I know, both are great universities and have top notch law schools. However, LSE appears to be perceived as the better one, especially for those who want to do commercial law and go into MC firms in the future. LSE seems to be harder to get into as well.
As for the teaching, I think you would get more information by doing a simple search at LSE and KCL forums on TSR. There were many threads made about this previously and most of them contain useful information. In terms of exams, you are likely to get a mixture of essay and problem questions at both.
Finally, have a look at the course content and the module choices, and see which suits you better. Both are in great locations and I doubt that there would be much difference in that respect.
Also, bear in mind that KCL has a larger intake then LSE, something like 200 students per year. However, LSE seems to have a more international student community then KCL.