This is just my opinion. In stead of trying to convince your mom into a very large bill that she may not be able to afford. Try doing this. You are 15 depending on your state you should be able to start working soon. Why don’t you sit down and talk to your mom. See if you pay for part of the braces would she then be able to yet you have them. She still may say no, at that point if you still want them you may have to wait until you are 18 to buy them for yourself.
I know that it is hard sometimes not to get what you want. However if you really want them you will make it happen even if you have to pay for them.