I have to say she's one lucky 13 yr old. My bum would of been tanned by my mother and father, and TONS of yelling.
The making her pay a bill is a good idea but what 13 gets that kind of allowance, I sure didn't, I didn't even get allowance, it was called pop bottles.
I would make her take it out, since it will ruin her teeth, and could get stuck in her braces. Since it was a home job, it will most likely get affected if the girl didn't use sterial glove, rubbing alcohol on the area, or even clean the earring so there was no bacteria. She could also chip a tooth on the pierce too.
Find pics of what a it will look like after it grows back. sometimes things like that will leave a scar. Like tongue rings can leave a indent after they grow back.
One other thing, if she' allergic to nickel in earrings, and can only ear gold, sterling silver, than she will most likely have a problem with the material that is in her chin.
For punshiment, its hard because at that age no kid really cares what they get tooken away, or least that was how my brother was. But I would contact that girl that did this to your daughter, parents, and tell them what the girl did.
Since most 13 year olds don't have cash to pay a bill, have her sit down and help you budget your bills, so she can see what adult life is like. Have help you plan the food menu for how ever long you shop for, and have her pick out the food and tally how much the food bill will be. Do a pretend apartment hunting, and add ALL the bills that will be included like car insurance, and so forth.
Another idea give her play money persay how a young 20 year old would have living on their own. Make her use her pretent money for each meal that she eats. Like a sandwich is 5.00, a glass of water is 1.25, pop is 2. and so forth. Do this for a week.
Or better yet, what I hated is when I did something like pierce my upper ear my mother did it. Go get a magnetic fake piercing and put one in our lip and walk around with it. Usually if a parent does it, the teen will hate it.
Those were just a few ideas. Sometimes teens make you so mad you don't or can't think of a good punishment. Good luck.