Well I think I can offer a few helpful tips, having done a Master's at the LSE myself.
I'm sorry about the funding problems. Unfortunately, many funding deadlines are very early in the year. Once the deadline is passed, there is not much you can do about it. If you still can't find funding, your best bet is to defer admission and start applying now for funding for the academic year 2012-3. Unfortunately there will always be more people needing funding than funds.
I wouldn't be too overwhelmed by the 'prestige' of the LSE. To be honest I disliked the place immensely. Their masters' programmes in particular are money-making machines. The school is a master's degree factory to be honest. And one of their most lucrative target audiences are starry-eyed Americans who love the idea of living in London. Sorry to sound harsh but unfortunately this is the truth.
Class sizes are large, and a lot of the course is actually self-study. Good luck trying to find a seat in the library; the campus is small and cramped and dingy and there is nowhere for students to hang out other than a cigarette-strewn alleyway and overpriced coffee places.
And I won't even get started on the ludicrous cost of living in London.
On a more positive note, being diabetic is completely irrelevant to your academics, to any funding decisions, and to any visa decisions. As a student on a legal visa, you are entitled to full NHS rights. You can see your GP as many times as you like for free, hospital care is free, and all your basic diabetes kit is free. You won't be able to get a pump or a CGMS but the insulin, needles, lancets, test strips and all the other basic kit you need to keep yourself alive is free. You will need to complete some easy paperwork but that's relatively painless.
It is an incredible relief not to have to worry about buying all the basic and vital diabetes kit. So that is one worry settled at least. Good luck on the hard part, the academic funding!