While it's true that different people look at money differently and some might consider $15 for a cake a lot, this woman knew that's not how much that cake cost. The first tip off is that she said, "I saved out $15 for you-that's how much it would have been at Costco." She was justifying why she was only paying $15 and trying to talk fast. Plus, I think there are VERY few people, even "cake civilians" who don't realize that a homemade cake is more expensive than a grocery store cake. I looked at your cake in the photos, and there is NO WAY that anyone, even a cheapskate, would have thought that that cake was $15 worth or work OR cake. It sounds like she overspent on everything else and was trying to recoup her expenses!
I agree that you should have clairified with her before, but I still think she had to know she was shorting you by quite a lot (even if she didn't know the full amount). I wouldn't do another cake for her again, and I think everyone else here is right, you should just send back the $15, with the "paid" invoice and tell the woman that it's now your gift. I wouldn't be rude, just state the facts and wish her daughter a happy birthday. If she has any sense at all, she'll be horrified.