You should be filing a motion for leave to file reply brief instanter. The motion is a request to be allowed to follow out if rule (after the prescribed time) explaining why your late and, in the interest of justice, asking them to hear you out anyway. The brief should be included. “Instanter” means that the brief is attached to the request.
But generally speaking, sur-reply papers are not permitted without leave of court. However, if there is a brand new affidavit submitted on reply it should either (1) not be considered by the court in support of motion; or (2) you can ask the court for permission to submit the sur-reply to address that affidavit only.