Start: Adelard Took
'I am simply at my wit's end! Nothing we have done has appeased the Old Took's ghost. I can think of nothing else to do. I will never have my library back. What to do, what to do....
'Wait...no, it's too terrible to ask! But I must have my library back. You have done so much for me already, but could I ask you to muster up your courage and face the Old Took's ghost yourself?
'With great caution, enter my library and perhaps -- since you have done so much to try and appease him -- the Old Took will grant you an audience. His presence seems to be strongest in the far corner of the room, by my small bookshelf. Do please be careful!'
Nothing you have done for Adelard Took has appeared to have any effect whatsoever on the "ghost" in Adelard's library.
1. The ghost's presence is strongest in the far corner of Adelard's library, near the small bookshelf.
After all else failed, Adelard Took has asked you to enter his library and confront the ghost of the Old Took.
2. Adelard Took is outside the library in the Great Smials.
In confronting the "ghost" in Adelard Took's library, you discovered that it was nothing more than a squirrel hiding behind the bookcase. You should tell Adelard that it is safe to return to his library.
At the end of the room sits a clickable bookcase. If you're unable to summon the squirrel, someone else may be doing the quest - so just wait it out. And even though it says to coax the squirrel out, you need to do nothing but wait for it to leave to get the update.
Sharpened Dagger OR Tookish Earring