UCL has a far better reputation than KCL and does better in the league tables but I've heard KCL is better known internationally? I go to KCL for my postgrad and it seems to impress people when they find out where I am, despite it's lower than expected league table rankings. I don't know if it's a postgrad thing or just my course but there are quite a few people with qualifications you would not expect on a law postgrad at KCL, e.g. people who got a 2.2 in law when the entry requirements say 2.1 or first. And there was me back in August last year, worrying that my high 2.1 wouldn't be good enough to get in! I have to say also the way KCL is run (maybe just my department but not from what I've heard) is a complete shambles. The tutors aren't all that great, and the whole thing is just a bit chaotic. From speaking to undergrads in my dept that's the same for them too.
It's a lovely uni though in terms of location and the actual buildings, there's a lot going on and the societies are good. I think you get more opportunities from being in London than you would do in regional unis. I went to a uni in the south for undergrad which is also usually not great in the league tables despite being a 'good' Russell Group uni, but it was run fantastically, the tutors were amazing, there was a lot of support etc - BUT London does have more opportunities. It depends what you're looking for in a uni I think