Does a wife become divorced after her husband says to her: "I swear by Allah the Almighty, if you repeat the insult, then you are divorced," and then she repeated insulting his mother before he finished speaking?

You made your wife's divorce contingent on her repeating the insult to your mother, and you confirmed this with an oath. She has now repeated the insult to your mother. Therefore, the ruling (fatwa) we follow is that the divorce has occurred, and this is the view of the majority of scholars. If this pronouncement of divorce does not complete the three divorces, then you may take your wife back during her 'iddah (waiting period). Some scholars, like Sheikh Al-Islam Ibn Taymiyyah, hold the view that if a divorce is made contingent on a condition with the intention of threat or emphasis, and not with the intention of effecting a divorce, then it takes the ruling of an oath. If the condition is violated, the one who swore the oath must pay an expiation for a broken oath (kaffarat yamin), and no divorce occurs.