There's often a general student perception that KCL is a little behind UCL and LSE in general. Perception or reality or a combination does result in LSE attracting the highest average entrant grades of the 3 universities for all subjects as a whole (195). UCL has 190 and KCL 171.
Yet in world rankings, for all subjects as a whole, UCL is ahead but next it's KCL and LSE is just one place behind KCL.
For Law, The Complete University Guide says that, of the 3 universities, average grades achieved by recent entrants have been highest at KCL (93%, same as Oxford), followed by LSE (92%, same as Cambridge), then UCL (91%, 5% ahead of the next university, Durham).
For research quality in Law, UCL is top (91%). LSE is in joint 8th place on 85%. KCL is in joint 19th position on 83%. Unfortunately, this doesn't tell us research intensity in Law - what percentage of staff in the dept are doing research. If there's just 1 great person doing any kind of a research at a university, it's going to unfairly make that university look better than a place that actually has even greater researchers but also some slightly worse researchers that results in the average quality % reducing. For the universities as a whole, across all subjects, UCL has the highest research intensity of the 3 universities (0.72), then KCL (0.69), then LSE (0.65).
For graduate prospects, 82% of UCL Law graduates went on to do something related to Law. For LSE it's 77%. For KCL, it's 76%.
I have a sense that LSE is for specialists but might not provide the most rounded, enjoyable, student experience.
UCL has approx 52.6% international students. LSE has approx 51.3% international students. KCL has approx 39.1% international students. These stats don't necessarily imply anything about the quality of the entrants but English might not be the language that some of them have most mastered and the qualifications they have might not be directly comparable to A-Levels.
Given that KCL is generally underrated by the general student compared to the others, yet has not only attracted Law students who achieve a slightly higher average grades but it's likely that more of their Law students are proficiently fluent in English, I do feel that KCL is probably genuinely the best of those for Law for most undergraduates, then UCL, then LSE.