This mission is very short and to the point: Give the Jarl of Whiterun Ulfric’s Axe and let him make up his mind on who he’s going to join. Head to Whiterun/Dragonreach and give the Jarl the axe. He’ll consort with his advisor’s for awhile and will finally decide to join the Imperials, giving you back the axe and writing a letter to the Imperials for aid.
Return to the Palace of the Kings and let Ulfric know. He will consort with Galmar and will decide the time has come to take Whiterun by force. He will tell you to meet Galmar at the camp outside of Whiterun for your next set of orders. This ends the current quest and starts the next quest: "Battle for Whiterun".