Did you drop the staff near the shrine?
Open the inventory and drop. On the PC it's a shift click in the inventory.
BTW, it just so happens I'm planning on doing this quest in a few hours anyway. If I see any other details I could post.
But not I recommend just going to where the shrine is and drop the staff.
EDIT:
I just did that quest now. After you get the quest update you say you have:
Acadian, you have summarized the problem correctly. I have been all around and on top of the alter and no dialog for dropping the staff.
Mirocu, My last entry is: "I've arrived at the Shrine to Sheogorath. I should find the alter and place the staff neat it." ...
'
All you need to do is drop the staff. There is no further update until you drop the staff. Also you don't activate anything, you just need to drop it out of inventory. Like I say for the PC that is shift-left click on the inventory item.
I'm guessing that your looking for something to activate or some other quest update to drop, but there isn't any - you just drop the staff. There is a non-quest pop up that say you no longer feel compelled to keep the staff, but that is not a journal update.