Contact the financial ombudsman, they will be able to help, and in a lot of cases they side with the consumer. http://www.financial-ombudsman.org.uk
I got cheap UK travel insurance, got my jaw broken in Australia, racked up huge hospital bills, hotel bills, curtailment bills...
They tried every trick in the book to not pay.
Although it took over a year the ombudsman made them repay every penny.
It seems you documented everything well - the ombudsman attaches a huge weighting to the medical opinions of treating doctors, so you have a good shot.