I went to an appointment, and they told me my power cable was the problem, which it was. So I handed over 50 big ones for a new cable and voila. Exactly a year after first buying it.
Hold on! Why were you charged. If the item failed within the warranty (assume you had to wait at least a day for your appointment) then it should have been replaced free of charge! Looks like they haven taken you for a mug there!
I think they can trace your purchase history via your Apple ID, if you have one of course.