They will duck the issue, they're Currys!!
The trouble is, this appliance is only sold through Currys. Even the service is arranged by them. Ring up Kenwood and they redirect you to Currys!
How much out of warranty is it?
European law says that all goods must have at least a 2 year warranty.
If it's older than that, you could quote chapter and verse from the Consumer Rights Act 2015 and see what they say.
CRA 2015 says that if the fault occurs after six months, the consumer has to prove that the problem was down to a fault or issue at the manufacturer, as opposed to wear and tear or misuse by the consumer.
In accordance with a law known as the Statutes of Limitations, consumers have this right for six years in England and five years in Scotland.
Basically, you may need to get an independent report from an appliance expert to confirm the fault and, importantly, where it originated.
If it turns out that there is an inherent manufacturers fault (and there is in the fridge door wiring loom for that Kenwood model!), Currys will have to assist you.
You could try telling Currys you are aware there is a recognised design fault with the door wiring loom as other customers with your fridge have had rectification work carried out FOC.
If they accept this, it may save you the expense of an independent report.
Either way, they cannot shirk their responsibilities, even though they have tried to shake you off.