The lights are on
Follow Irileth to the Western Watchtower, which was destroyed by Mirmulnir. The battle was lead by Jarl Balgruuf the Greater's housecarl, Irileth. Many men from the Whiterun garrison accompany the Dovahkiin and Irileth in battle with the Dragon, equipped with bows and swords.
Stay hidden by the debris of the wrecked tower, or climb to the tower's summit. From either location, use ranged weaponry or spells to attack Mirmulnir. Irileth and the guards may be able to distract Mirmulnir, long enough for a melee-oriented Dragonborn to attack from the flank or behind. Beware of his deadly breath.
After Mirmulnir was defeated, the Dovahkiin absorbed his soul, much to the surprise and astonishment of the soldiers. The soldiers proclaim that he must be the Dovahkiin. Irileth, who is skeptical about the theory, told the Dragonborn to report back to the Jarl. On their way back, the Dovahkiin heard the call of the Greybeards to High Hrothgar. Jarl Balgruuf the Greater explains what it means to be a dragonborn to the Dovahkiin.
The Jarl promotes the Dovahkiin to Thane of Whiterun, gives them the Axe of Whiterun, and assigns Lydia as their housecarl.