Skyrim:Dungeon Delving (Jarl - Caves)

Quick Walkthrough
# Obtain quest from Korir, the Jarl of Winterhold.
# Journey to the target location.
# Retrieve the Helm of Winterhold.
# Return to Korir with the Helm.

Detailed Walkthrough
When asking the Jarl of Winterhold, Korir, for work, you may be asked to find the missing Helm of Winterhold, which was worn by   in the First Era:

:'''You're the Jarl? I'm looking for work.
:"Winterhold has lost much of its history, and with that much of its power. I aim to change that. I've heard rumors of the resting place of the Helm of Winterhold, the very same helm that Jarl Hanse wore in the First Era. Hanse was in line to be High King of Skyrim, you know. Having that might get the ear of the other Holds and give me some authority."''

Once you've agreed to find it, head to the location marked on your map, and head to the boss chest, where you will find the Helm. The location will be randomly selected from the following:  .

After you have retrieved the helm, return to Korir to receive your reward.
 

Notes
The journal entry for Driftshade Refuge is listed as Driftshade Sanctuary.
If the location of the helm is at Hela's folly the helm will need be stolen from a locked chest. If caught by Deekus he will take everything stolen from the chest except the helm. If arrested by a guard they will also not take the helm as it's status as a quest item supercedes its stolen status.
You cannot get this quest if Korir is not Jarl of Winterhold. Fortunately it is only a prerequisite for becoming Thane while Korir is Jarl: if he is replaced, his replacement Kraldar will give you the title without completing this quest or any other.
The Helm can be pickpocketed back off Korir if you are skilled enough. It will no longer be a Quest Item, but appears to retain its Stolen status even if sold to a Thieves Guild fence and repurchased (at least, this was the case with Mallus Maccius in Skyrim Special Edition).
Unlike many "named" objects, there is nothing much special about the Helm of Winterhold. Once it stops being a Quest Item, it has the exact same properties and sale price as an unenchanted Steel Helmet.
As of the latest version of Skyrim Special Edition, this also means it can be tempered with a Steel Ingot and the Steel Smithing perk. It can even be enchanted. The latter is not possible while it is still a quest item.

Bugs
 
 
 
 
The quest can be completed by attacking and luring Korir outside, calming him, and then speaking with him before he heads back inside. This will incur a bounty. Trying to engage him in conversation just as you are leaving will also lead to him following you outside.
 Alternatively, open up the console, click on the jarl and then type   into the box. This adds an NPC to the player ally faction, which will make a normally hostile NPC be non-hostile. This should allow you to complete the quest.
 You can also finish the quest by typing  . This will also remove the helmet from your inventory.
 
 Some reports suggest that the quest is flagged as completed as soon as one gains access to the College.
 This quest can only be started the first time you visit Winterhold, if you fast travel away before initiating the quest then you won't be able to start it.
 Starting the quest via console does not work. You can bypass this quest and still become Thane of Winterhold on PC with  .

Quest Stages