Must I repeat the prayer, or perform the forgotten rak'ah after remembering it, having exited the prayer and walked in the room for a few seconds without speaking?

If you forget a rak'ah and then remember it and perform it after a short interval, your prayer is valid, because you rectified it "shortly before a long separation occurred." The determination of what constitutes a long or short separation is referred to custom and common practice. As for walking and not facing the Qiblah in this situation, it does not invalidate the prayer, because you did so believing your prayer was complete. And performing actions that contradict the prayer based on the assumption of its completion is excused.