What is the specified duration for making up a missed prayer with a legitimate excuse after remembering it, such that making it up after this period would not be beneficial, and the person would be considered to have intentionally abandoned it, requiring repentance? And does this duration equal the original prayer time?

Prayer is a timed act of worship, and its time must be observed. Whoever misses a prayer must make it up immediately once the impediment is removed, based on the Prophet's (peace and blessings be upon him) saying: "Whoever forgets a prayer, let him pray it when he remembers it, for there is no expiation for it except that." It must be made up immediately to absolve one's responsibility, and the obligation to make it up is not waived by delay. Whoever delays making it up without an excuse incurs sin. Delay is permissible for an excuse such as essential eating, drinking, or sleeping, or fulfilling an unavoidable necessity.