I'd also bear in mind the student experience, LSE is a very different enviroment to KCL. LSE is incredibly focused, and based on what others have said, often the teaching quality is incredibly poor and the social life isn't the best.
LSE will give you better prospects, but, as many will tell you, after 3 months of work the university you went to starts to not matter a jot. However, LSE is undoubtably better and depending on what you'd like to do could benefit you significantly.