Is the questioner considered a severer of kinship ties and disobedient to his mother for refusing to reconcile with his sister, who has been described as ill-mannered, a troublemaker, and one who commits forbidden acts such as fornication and abortion, and for which the questioner possesses conclusive proof of her behavior, despite his mother's pressure on him to reconcile, and his condition of her sincere repentance to reverse his decision?

What you described your sister as having—bad manners, spreading gossip, slandering others, causing discord among people, disobeying her mother and siblings, having marital problems, being disobedient to her husband, traveling to infidel countries, and conceiving out of wedlock—all these matters are far from uprightness. However, maintaining kinship ties (silat al-rahim) is one of the most emphasized obligations, and Allah and His Messenger (peace be upon him) have commanded it, even with a disbeliever. It is therefore even more essential with a disobedient Muslim. Cutting off kinship ties from one side does not permit cutting them off from the other side.

Based on this, and given your duty to obey your mother in what is right, you must reconcile with your sister to the extent that kinship ties are maintained, while making it clear that you disapprove of her behavior. If you fear harm from excessive interaction with her, then limit your connection to the minimum that satisfies your mother. Part of maintaining kinship ties is to pray for her guidance, advise her, remind her of Allah, and warn her of the seriousness of her situation. If she is guided, that is the desired outcome; otherwise, you would have fulfilled your duty and will not be described as disobedient or as having severed kinship ties.