Is the questioner considered sinful for her endeavor to end her brother's relationship with a girl who caused her harm, and is her action of informing her brother about the girl's derogatory remarks about him considered backbiting, knowing that this led to the termination of the relationship?

Tale-bearing (namima) is forbidden and a major sin. It is not permissible except out of necessity that cannot be averted otherwise. If what you informed your brother about was evil that the girl wrote about him, then this is not permissible because it is treating evil with similar evil, and there is no necessity in it. It is not permissible to avert harm from oneself by harming others. However, if the speech contains harm that cannot be avoided except by informing the brother about it, then this is permissible.