I qualified under the older LPC & Training contract route so I'm a bit less familiar with the SQE (so plz other posters jump in to correct me as needed) - The SQE is the practical step of your legal education, you get a masters along the way but law firms
won't be looking at that masters as upgrading or replacing your undegrad degree classification, the LLM part is really just a marketing gimmick by the course providers - the key bit of that course is passing the SQE exams themselves. In my day when I completed the law conversion and the LPC I got a "bonus" LLB from the College/University of law, which *no-one*(!) was ever interested in. Of course if that LLM element is key to secure your ongoing UK visa status then that's important and a clear reason to take the course - but it won't improve the academic side of your application.
You will need to do the SQE to qualify as a solicitor so its a clear next step in one sense, but it doesn't make you more marketable as a candidates because it's simply a course that all solicitors will be doing (if they don't do LPC or some of the other more niche routes for non-grads). If you aren't able to secure your 2 years training with a law firm it's a lot of money to spend, without any guarantee of that legal career at the end of it - if you have the money and can take that risk - then it's an option.
Have your careers dept at KCL offered any suggestions or guidance on next steps in law? Have have you applied for any work experience or vac schemes with law firms in London to date? - If you're successful in obtaining competitive work experience placements in law that's usually a good guide that you're in with a good chance of a training contract.
https://www.chambersstudent.co.uk/law-firms/getting-a-training-contract/application-and-selection-criteriaThe chambers link above shows how tough it is re the number of applicants to training contracts, obviously many candidates will be making 10-20 (or more) training contract applications so it's not quite as bad as the numbers suggest, but the point is this a very competitive graduate career to enter.
It's going to be tough with the 2.2, really tough. Not to say it's impossible, but you're going to be up against lots of your peers with 2.1s, some with firsts (but important to stress a 1st is *not* needed at all) and lots of solid work experience on the CV. For all but the super exceptional candidates who can afford to be picky it's less of where you can gain the "best" training" (which is really impossible to say anyway), and more "where can I obtain a commercial training contract". You need to obtain solid work experience in the UK, apply for vacation schemes, open days etc... any links through KCL careers support - apply apply apply. Can you paralegal at a commercial firm in London for a year or two before looking to start any further SQE studies? If you can - apply apply apply.