Do you ever think about how desperate Ziva was to see the good in Ari? How unwilling she was to believe that Ari could have possibly done all of the things that he was accused of doing. He was the only sibling she had left, if not one of the most important people in her life. She tried so hard to find every reason why it couldn't be him. She was so afraid of losing him and she tells him so during that brief conversation.
And when she finally realizes that Ari is guilty of all that he's been accused of, that he's become a monster in his own way, it breaks her. Ziva can't turn a blind eye anymore. The brother she wanted to protect is gone and she must follow through with her orders no matter how much it will destroy her to do so.
Ziva David was given orders to kill her brother. Yes. But in that moment it didn't feel like she was killing her brother instead killing the monster that he had become.