Win the lottery?
Without getting into debt I can't think of a way to find �6k. And if you're 14 you can't get a bank loan.
I'm sure there's a question of responsibility for supervising children, but at the end of the day you're 14, you knew that what you were doing had risks, and accidents happen.
I'm really sorry to hear about your tooth. I've had tracks in my youth and I know how it felt when they came off. Mind you, since then I've chipped my front tooth twice, once on a beer bottle, once on the bottom of a swimming pool. And my back tooth got chipped on a piece of glass in a muffin. I'm not sure I can sue anyone for any of those things though. In one case I ate the evidence :-)
Perhaps your mum can avoid horrible litigation by arranging some kind of loan? I'm sure she would find the idea of you trying to raise six grand ridiculous. If she's of a mind to sue, you trying to raise the money might not stop her.
Get a second opinion on the cost. Was this an NHS dentist?
All the best.