Hiya
I'm also planning on applying to Oxford and KCL this year as well as Warwick, Southampton and Leicester. I would say like the others have said already, try discussing politics in relation law, maybe sort of discuss how politics has affected the ability to ensure that laws are effective (this being said don't make it too opinionated either as I've heard this can sometimes be a disadvantage to applications.)
LSE is really competitive but if you're looking for other unis in London that are still pretty good but not as competitive, I would recommend SOAS and Brunel- from what I've heard they're pretty good. Also you could consider Kent and Surrey too, although they're not in London they're both still pretty close to the city.