To investigate the Royal Scribe’s office in End of The Snake in Assassin’s Creed: Origins interact with the scroll on the desk. Then interact with the broken ladder on the floor. Climb to the top of the shelves at the back right of the room:
Then go along the top all the way to the left, to the statue and then the shelves on the left of the Royal Scribe’s office. Pick up the key at the top of the left shelves:
Use the key to interact with the chest.