i've been to open days for both, i'm a law applicant, and well...
both open days, the current students mentioned the slight rivalry, mostly over jeremy betham's head.
it is down to preference i suppose.
both are relatively near to each other, with lse in between.
however, king's is much more city centred, whereas ucl is much quieter and campus like... although the campuses of king's are all within short walking distances.