Just ask the judge what is the client's damage

First, this goes under contract law. Contract law is about business, is about making someone whole after suffering some damage. It is rarely used to "punish" someone else for breach the contract.

Second, tell the judge what the contract is, even it is oral and not written. "If I fixed it, pay me; if I can't, it's free and you take it somewhere else". Tell judge no part of the contract have you breached. Don't respond to the other party's accusation. His email is highly amateur and you don't have to respond.

If you have to respond, you can say the client is not making a legal argument, but a list of assumed facts that you can't respond to.  (such as nobody else will repair it since you touched it, ask the client find ANY repair person to the court and testify). 

Provide evidence that this is the understanding ever since from the start. Next time you should let people sign this kind of contract, not just a liability waiver. （免责条款）.

Third, even if you breached the contract, ask the judge where is the client's damage? If there is no damage, there is no way you should pay for anything.