I don't think there's any reason to be shy about it. If you walked in and explained what had happened, then made your dd hand over the cash they're only going to think that you're being a fantastic parent, unless they're really dim, that is.
Shall I tell you what happened to me and my dd?
When she was 7, she went with a friend to the friend's dad's workplace, a bookshop. They were left unsupervised for a little while and played a game that involved them defacing about £400 worth of hardback books
After discussing it with the friend's dad, he decided that the best thing would be for him to pay back the money by doing overtime. He didn't ask us for money because he shouldn't have left them unsupervised, and also he knew that we were skint at the time (dh was a student then).
There was no way on earth I was going to let dd get away with it, I had to ram home the message that defacing other people's property (especially to that extent) wasn't on.
I made her write an apology letter, then frogmarched her down to the shop to give it to the manager by hand.
Yes, I was embarrassed, but not half as embarrassed as dd was. The manager appreciated it, had a chat with dd about what had happened but wasn't horrible to her.
She's never done it again, I can tell you!