Is my local Jaguar dealer liable for the repairs costs of my F-Type?

I have a 2013 Jaguar F-Type, which I have owned from new. The car developed a misfire and was transported back to the main dealer by the AA. The dealer diagnosed two spark plug failures and changed all six. 20 miles later the misfire re-appeared and the garage diagnosed an injector problem on cylinder four, but they snapped the injector off in the cylinder head by using a slide hammer to remove it. They are now quoting me £6000+ to replace the cylinder head and all ancillaries. I have told them I hold them responsible for the additional problem but to go ahead and to keep me informed. They know I will seek redress at a later stage. My internet research indicates that injector failure on the F-Type is not uncommon and that using an unapproved force to remove it will likely result in a snapped injector. I have asked the dealer if Jaguar will show goodwill to a valued customer and defray some of the costs. They said they would try. Are they liable for the additional costs including the charge for an incorrect diagnosis and six plugs which did not need to be replaced? What should I do?
If anyone is liable for this the garage is; definitely not JLR unless JLR choses to intervene with some financial help for the dealers. This FAQ answer is more about rejecting cars or getting fault cars fixed or replaced, but the law is pretty much the same:
Answered by Honest John on

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