The way I see it, LSE, KCL and UCL are all so close to each other in terms of the rep of their law departments that it makes no significant difference which one you pick. Employers are shrewd enough to know that they are all amazing institutions and hard to get into, so appreciate them all. They're also not naive enough to think that just because you went to one of them, you're a fantastic applicant. It comes down to how you sell yourself afterwards and the skills you display. To that effect, choose the university that you personally could see yourself excelling at and are likely to gain the most from. UCL has smaller student body for law, and more intimate tutor groups. I've heard that King's offers more regular tutor sessions though (I can't say I know this for sure), so I suppose if you like that sort of guidance, then that's useful. Also King's recently moved into the East Wing of Somerset house, which looks bloody fantastic. As the home of the School of Law, you should expect to be spending some time there. If you think that sort of atmosphere would help you excel, then take that into account too.
In essence, what I'm saying is that the ranking differences are more or less negligible. They're volatile and the differences are tiny. What's more important is which course and institution suits you. I'm sorry I can't give you many specifics on the things you asked for, but I just thought I should point this out after seeing half the above posts harping on about rankings.