I sympathize with your case. A 2.1 is really not bad. You shouldn't denigrate yourself and view your peripatetic interests as requiring redemption.
Taking on a big unsolved problem without guidance or the background of a PhD student seems doomed to fail. Locking yourself in a library with all the world's books is unlikely to produce anything of merit. I have never heard of a case of a student producing something of "intriguing/charming/advanced" and using that to gain graduate admission. The romantic, amateur heroic view of math is largely bunk as pointed out by Terry Tao.
There is still hope. I know that undergraduate research is less common in the UK, but I would expect that if you email lots of professors in areas of interest to you and basically offer yourself as cheap or free labor (undergrad student level), there is a good chance that you'll be taken on as an unofficial research student by someone. I know many cases of people in math and science using this sort of informal contact to start research projects that eventually develop into PhD positions. Making yourself known to a tenured professor who can write you a strong recommendation is probably enough to get you a PhD position somewhere (in the US, UK or Europe). It is unlikely that claiming to solve a big problem or do research on your own is going to be trusted by graduate committees. You need recommendations from people trusted in the academic community.
There are MO users who have taken a decade or more off from education and successfully started PhD positions at Princeton and other top research institues. Good luck.
