The person you must save in this mission can be found on the higher floor of the mansion. You must set a new starting position to have more time to act.
Go directly to the nearby room after restarting the day.
There you must use the code you obtained to reach the previously unavailable area of the mansion.
Walk through the corridor in front of you. You shouldn't encounter anyone.
Interact with the area next to the stairs to go to the higher floor.
Once upstairs, use the door marked on the above picture.
You will enter the corridor shown above. Use the marked door to enter the nearby room.
Inside you will find another clock key. Go back to the corridor.
On the corridor you must go to the door shown on the picture.
You will find yourself near a bell. Use the exit to reach the place where you will continue your actions.
In the location first you must fix the clock and then synchronize it with your pocket watch by using the Z key. If you are sure you changed your starting position, press F1 once you're ready and start the day again.