What should I do to perform the Dhuhr prayer and help my exhausted mother at the same time without displeasing either of them?

There is no significant conflict between prayer and helping one's mother. A Muslim can prepare for prayer, perform it, and then hasten to help his mother. If the mother requests help before the prayer, the son should help her, unless the prayer time is short. If the prayer time becomes short, it is obligatory to begin the prayer, and delaying it is forbidden. There is no sin upon the son if the mother becomes angry in this situation, because there is no obedience to a creature in disobedience to the Creator. After the prayer, the son returns to help his mother, thereby pleasing his Lord and assisting his mother, while gently and wisely trying to explain the matter to his mother.