What is the ruling of Islamic law regarding a wife who abandons the marital home, takes its furniture and her daughter, and refuses reconciliation and a divorce by Khul'?

What the wife did is considered nushuz (disobedience), for which she is sinful, and she forfeits her right to maintenance and housing. She must return to her husband's home and obey him, for the Prophet, peace and blessings be upon him, said: "If I were to order anyone to prostrate to another, I would have ordered the wife to prostrate to her husband."

The husband must pursue means of reconciliation with her, such as sending two arbitrators from their families, as Allah Almighty says: "And if you fear dissension between the two, send an arbitrator from his people and an arbitrator from her people." Or, they should refer the matter to a Sharia judge to rule between them regarding the nushuz, custody, and household furnishings.