Is the saying: "Prayers upon the Prophet (peace be upon him) untie knots and relieve distress," permissible, and is its attribution to Ibn al-Qayyim correct?

This statement is correct Islamically and proven empirically. Making Salat on the Prophet (peace be upon him) a substitute for supplication brings good and repels evil, as per the Hadith of Ubayy ibn Ka'b (may Allah be pleased with him): "Then your worries will be sufficient for you, and your sins will be forgiven." Therefore, Salat on the Prophet (peace be upon him) in times of worries and hardships, and seeking forgiveness, suffices what concerns a person in the matters of their worldly life and the Hereafter.