Did the confirmation letter state that they would refuse further service? If not, they cannot rely on an email after the subsequent failure. They may have a clause in the contract itself to support them. However, I suspect a lawyer would say that by accepting further premiums, and
without a letter stating further breakdowns would not be repaired, they are obliged to provide that service.
Assuming the letter did not make the relevant statement, my next move would be to ask them to provide a copy of the clause upon which they are relying. Don't let them get away with it. Don't do the common thing, threatening to tell the newspapers. They can handle newspapers, even of they were interested.
If they cannot come up with the clause, ask them how to complain to the FCA. They will definitely not like that. It costs them money even if a complaint is not upheld.
I had a slightly similar issue with an insurance company. To cut a long story short, they refused a claim upon the definition of "property". After countless calls and discussions, I asked them to outline the clause or paragraph that would support their definition. They gave up and paid out.
Good luck.