Firstly, yes, the programme does exist. It is called Law & American Law LLB and JD. You do the first two years at King's and if you're one of the best students there (ie. predicted a First) you may have the chance to apply for one of two places. You have to have proof of the appropriate financial assets in order for you to be considered, which including living costs in New York City is around $90,000 per year. Moreover, students are placed in the second year of the JD programme and often have difficulty adopting the new lifestyle and academics. All in all, it is a graduate programme and you'll be studying with 25 year olds already holding a Bachelor's degree. If you think you're up for it you should try and see if you can make it even as far as King's prestigious LLB. While getting both a LLB from King's and a JD from Columbia Law School is an extraordinary achievement in 4 years, as it would take someone usually 10 years to do so, it is definitely not for everyone.
To address your queries for law and admission to King's, you need a score of 26 at least on the LNAT for a competitive application, a unique personal statement and although they do not specify any subject requirements, English Literature and History are the norm. If you plan to apply to the LSE you should probably steer away from the Law A Level.
I hope this helps. Good luck.