Click here for more info on the East Empire Blessings and how convincing the Jarls works.
The following options are available for both Ulfric and Brunwulf:
Regardless of who the Jarl of Winterhold is, a note can be found in the War Room.

Reading this note will give the player an optional miscellaneous objective to kill the spirit haunting Anga's Mill.
If the player heads to the mill at night, they will find a "Drowned Mage" spirit. If the player kills and loots the spirit, they will find a "Scribbled Note". Looting this will give the player an objective to read it.
Reading this will give the player an objective to find a ring in Yngol Barrow.
Upon looting the ring from a body in Yngol Barrow, the objective will update to tell the player to return it to its resting place.
Placing the ring on the body the objective points to will update the objective to prompt the player to tell the Jarl. Doing so will succeed the Favor.