Battle for Windhelm

Background
" I am to join General Tullius in the final attack on Windhelm, putting an end to the Stormcloak rebellion, once and for all. "

Walkthrough
The Battle for Windhelm is the final Imperial Legion quest, ordered by General Tullius, who wishes to take over the city of Windhelm to find Ulfric Stormcloak and force him to submit. After an inspirational speech by General Tullius, you fight through waves of Stormcloak defenders along with the legionnaires. The Palace of Kings is then raided, where Ulfric and his right-hand man Galmar Stone-Fist await you, the enemy. Once Ulfric is defeated, he will request that the player be the one to kill him, as it would "Make for a better song," although you can refuse and allow General Tullius to strike the final blow.

After Ulfric and Galmar are defeated, General Tullius will reward you with a leveled sword called the Elven sword of Leeching. Upon retuning to Windhelm, Brunwolf Free-Winter will become the Jarl and will offer you the title Thane of Eastmarch as well as the opportunity to buy Hjerim, the player house availabe in Windhelm (12,000). At this point the city is now Imperial, thus the Stormcloak guards are replaced by Imperial Soldiers.

Edit:

General Tullius will reward you with a sword, depend on your level what kind of sword he will give you. I got Ebony sword of the vampire.

Bugs

 * Sometimes during the battle with Ulfric and Galmar, both enemies are still tagged as essential, and both of your allies just kind of stand around after making their pre-fight speech. If this happens, you are stuck in the Palace of Kings with two unkillable enemies and two NPCs. Loading an earlier save does not prevent this from happening, and the console doesn't help either. There is no known fix for this bug.
 * Ulfric and Galmar's bodies may not disappear from the Palace of Kings, even when Brunwulf Free-Winter is Jarl.
 * Sometimes you will not recieve General Tulliuis' sword, you may still finish the quest but you will not recieve the sword. (Confirmed PS3)

Refrences
Video walkthrough