I just graduated from Warwick (BSc Management). I have no knowledge of Kings for a comparison, so I can only give you my experience. Warwick Business School (WBS) is almost a separate institution from Warwick University. (Think Harvard & Harvard Business School). WBS is excellently facilitated, especially with the new business school built specifically for undergrads. In terms of career prospects, we are a target university for the top financial firms in the world (Mckinsey, Bain, BCG, PWC, Deloitte, etc.). According to the grad diary, Warwick Business School grads earn the
3rd highest salary in the country after graduating 5 years after employment. See again how this table separates WBS (ranked 3rd) and Warwick University (rank 15th). Kings did not make it to this list.
Personally speaking, I really enjoyed the state-of-the-art facilities and infrastructure at WBS, in addition to the flexibility in module choice. For instance, each year we get 10 modules to choose from. Management students have something like 1 or 2 core modules in their 2nd and 3rd year, and the other 8 are elective. This gave me incredible flexibility in what I wanted to study, and I ended up studying Law, Accounting, Economics, Marketing, Taxation, Finance in New Ventures, etc. I really enjoyed this freedom, and the spice of variety I believe is what kept me interested, and therefore kept me working hard enough to have graduated with a 1st class degree.
Another interesting point that I've heard, is that Warwick is a university that's up there with the top 5 universities in the country, without the price tag of living in London / Oxford / Cambridge. I've heard that it's the most economical degree, in terms of cost-of-living vs pedigree. Perhaps something to think about. That said, WBS own a floor of The Shard in London, and so you're welcome to go for the free talks held there. However that's completely voluntary, and it has nothing to do with your degree. I went to a few events just for my interest in Consulting as a career.
It won't come as a surprise that I'd advise WBS over Kings. In my 3 years at Warwick I haven't heard of Kings as a competitor to us. In terms of the general feel of pedigree, we're below Oxbridge (obviously), above Durham, and about on par with Bath.
That said, due to Warwick's international reputation, there's a lot of diversity in the business school. That can be a put off to some domestic students.
Good luck