Is it permissible to set a time limit for paying the institute director's commission for each student he brings, given that the previous agreement had no specified duration?

This transaction is a ju'alah (a contract for a specific undertaking), which is a permissible but non-binding contract. It only becomes binding after the completion of the work. Therefore, there is no issue in canceling or modifying the agreement. A ju'alah is a permissible contract for both parties to retract before the work is accomplished. If the one who commissioned the ju'alah retracts before the work is begun, then he owes nothing. If he retracts after the work has commenced, then he owes the worker a fair wage (ajr al-mithl). If the worker cancels before completing the work, then he is owed nothing.