What is the ruling of the Sharia on the return of a sister to her father's house despite his refusal of her return, and is it obligatory for me to do so to mend family relations?

What your sister has fallen into is a great evil, and she is the first to bear the pain and shame. The first one to abandon her is the one who deceived her, as Ibn Hazm said: "The first one to deem the adulteress worthless in his eyes is the one who commits adultery with her!!" This resembles the act of Satan, who whispers disbelief to a person and then disavows him, as in the Almighty's saying: (Like the example of Satan when he said to man, "Disbelieve." But when he disbelieved, he said, "Indeed, I am disassociated from you. Indeed, I fear Allah, Lord of the worlds." So the outcome for both of them was that they would be in the Fire, abiding eternally therein. And that is the recompense of the wrongdoers) Al-Hashr/16-17.

It is not wise for the guardian to confront this calamity with emotional outburst. Expelling the girl or killing her only worsens the matter and may cause her to fall into the snares of immorality and seduction. Wisdom lies in closing the doors to corruption, rectifying her situation, and purifying her with sincere repentance and uprightness, along with disciplining and caring for her. The Prophet, peace and blessings be upon him, commanded showing kindness to the repentant woman, as he did with Al-Ghamidiyyah, when he said to her guardian: "Treat her well, and when she delivers, bring her to me." Expelling the girl contradicts the duty of care and kindness. Therefore, the guardian must not remove his daughter from his care, but rather increase his supervision and diligence in closing the doors of evil to her.