Hello and thanks for using Just Answer.
My name is ***** ***** am happy to assist you with your enquiry.
Please be rest assured that you are not liable to pay any charge and certainly don't send them anything other than the £2.20!. Although these private Companies threaten debt collection and Court action, they are never successful in any such proceedings provided the other party defends matters and nor are they legally entitled to demand any payment from you even if you have contravened the parking regulations.
From what you say, you have not even contravened the parking regulations in the car park, and you therefore need to write to them to appeal the fine, setting out the nature of your visit and the non working of the machine.
All private car park charges are based on Contract law- for a penalty to become payable, there has to be a contract between the parties and it will be impossible for them to show such a Contract existed.
I attach a link to the Money Saving Expert website, which has plenty of information on how to appeal.
I hope this assists and sets out the basic legal position.
These Companies rely on the public to pay their fines, and sadly the majority of people do end up paying the fine out of ignorance of the law.
God Luck and please could you rate my Answer.