Is a wife's departure from her husband's home to her father's home without his permission, while she is determined to divorce, considered a divorce? And does her waiting period (iddah) expire and she becomes irrevocably divorced (ba'in baynunah kubra) by her staying in her father's house for three menstrual cycles, despite her refusal to return even though the husband, who used to beat her frequently, has repented from that?

A wife's mere departure from her husband's home without his permission does not constitute divorce. Divorce is enacted by the husband or by a judge's ruling. Her leaving without his permission is an act of disobedience and insubordination (nushooz), which forfeits her right to maintenance. If you have not divorced her, you should try to reach an understanding with her and mend the situation. If this does not work, then two arbitrators should intervene, one from your family and one from hers, either for reconciliation or for divorce if reconciliation is not possible.