After climbing up the grappling hook and over the balcony, you will be in an area with a bunch of plants. Do the following:
Open the door to the study (left of the tree with sap on it):
Panel piece - In the plant area, spin around until you see a fallen tree. Pick this up from underneath it.
Spin around until you find the door missing the handle. Use the brass handle on the door to open the study.
Use the panel piece on the puzzle next to the door in the right side of the study room. Complete the puzzle.
ACHIEVEMENT UNLOCKED when you enter the library room.