Why won't Korir offer me work (retrieving the Helm of Winterhold)?
The discussion page for this quest on UESP discusses this issue. It seems to be related to the Hearthfire addon.
If you have Hearthfire installed and have any one of the of "Build your own Home quests" started but not completed, you will not be allowed to accept any Thane quests from any other Thanes in any other holds. Even though I had already become a Thane in Hjaalmarch, I had not yet visited the homestead property, nor began construction on the house. Upon visiting the property and building the foundation, the quest completed. I immediately returned to Winterhold and was able to begin and instantly finish this quest.
So if you have Hearthfire installed, try to visit your property and build the foundation. The quest should complete and you'll be able to take "Become Thane of X" quests.
Becoming Thane of Winterhold is not exclusive to Korir. Additionally, the Helm quest is a technically radiant quest and doesn't have easily accessible quest triggers.
To ensure all conditions are met to get Korir to talk business, ensure the following are all true:
- It is during Korir's normal court hours. Target 10AM-1PM and you're probably good.
- Korir is not under attack
- You did not tell Korir "no" if he offered you the task previously.
If you shut Korir down on his request the first time, or broke dialog before he could offer it, it's possible his dialog bugged and he considers it a rejection of the task. If so, you can either reload a previous save before you met him, or win the war for the Empire.
If the Empire wins Winterhold, via war or just trade in Season Unending, they seat Kraldar as Jarl. Kraldar will give you the Thane quest of just helping people in the hold, instead of sending you to get the Helm first.