The first thing I'd say is don't give up! Getting a training contract is still possible but be prepared for the fact that you might struggle initially.
I was in a similar position to you when I finished my LPC many years ago. What really helped boost my applications was the post law school experience I built up on my cv. It took me just over two years, but hard work, good experience, and persistence, really did pay off in the end. Are you set on training in private practice or would you consider the public sector? I eventually got my TC with a local authority outside London, and my first qualified job was in-house (private sector) back in London. LA pay isn't necessarily as good but the experience was invaluable, and more importantly, I qualified! Also, think about applying to the Government Legal Service (GLS) and medium sized law firms for a training contract, if you're keen to stay in London.
An LLM would never be wasted. If it's something you want to do. Especially if your final grade shows steady progression from your LLB. I think some LPC providers offer the option of converting your LPC to an LLM by writing a dissertation!? A friend of mine opted to do that at College of Law (York) several years ago. So you might want to look into that as an option.
Try and get as much work experience as possible. If you decide to do postgrad, apply for vacation schemes. Maybe talk to a few recruitment consultants about any holiday placements!? They may be able to advise you. Once you've completed your LPC. If you haven't secured a TC, then apply for paralegal jobs. Some can even lead on to a TC, just don't rely on that happening. Keep applying.
I hope all goes well