What is the ruling of Islamic law regarding how to settle a debt owed by a Mudarib partner due to his negligence in managing funds, especially since the other partner is demanding the full amount or large installments that the Mudarib cannot fulfill from his salary, which covers his family's and mother's expenses?

You erred in what you did, and you must return the money to its owner, either in installments or otherwise. Your friend should be patient with you, for Allah the Almighty says: "And if there is one in [financial] hardship, then [let there be] postponement until [a time of] ease. But if you give [it in charity] as a charity, it is better for you, if you only knew." As long as you intend to repay, and your delay is due to an excuse, there is no sin upon you, even if death overtakes you before the full repayment. This is based on the Prophet Muhammad's (peace be upon him) saying: "Whoever takes people's money with the intention of repaying it, Allah will repay it on his behalf; and whoever takes it with the intention of destroying it, Allah will destroy him."