I am a foreign student and allow me to say that King's has a wonderful reputation in Asia and North America, which I why I opted to undertake study here.
I am not clued up on RHUL, but if you want market recognition, which we all do, then KCL is one of the crown jewels in the UK higher education scene. After Oxbridge, the only UK unis which are well received globally from my experience over 2 continets (Asia/North America) are King's, UCL, Imperial, LSE, SOAS and Manchester to a less degree.