I think that's perfect.
It's professional, without being rude or aggressive. However, it also lets her know you're not willing to be a pushover (especially when you've done nothing wrong).
It's very easy for me to be so confident ... after all, it's not my complaint.
When I had the complaint I felt sick and tearful for about 2 days until it was sorted. My customer did reply to the email I sent, telling her I couldn't investigate further - she said "I understand, but I won't be recommending you to anyone." Well, let's be honest. WHO in their right mind is honestly going to say to friends "I had a cake from xyz company... It was dry as cardboard but it's OK, because she gave me some money back... You should use order from them, too". NOT GOING TO HAPPEN. In reality, it would have been "I ordered a cake from xyz company and had to complain and get some money back because it was really dry." Who's going to place an order after a recommendation like that, eh?
Hope it all goes away now and you can carry on with nicer customers!