The quest begins while you are still speaking with Farengar. When you ask him about the reward for retrieving the Dragonstone, he directs you to the Jarl or his steward, Proventus Avenicci. The first quest objective is activated.

Talk to Jarl Balgruuf
You head over to the Jarl, to whom a guard is currently reporting a dragon attack. The dragon is said to have come from the south and was circling above the western tower. The Jarl tasks his huscarl, the Dark Elf Irileth, with investigating the incident and wants the player to accompany her. After all, you survived the dragon attack in Helgen and possess relevant experience.
Following this, he rewards the player for completing the task for Farengar. While Irileth gathers a few soldiers and heads out ahead, you can take your time to explore Dragonsreach and the city.

Meet Irileth near the Western Watchtower
You leave the city heading west and join the waiting squad. Irileth is concerned because the Western Watchtower has already been attacked. She orders everyone to look around and search for survivors to gather more information about what happened. Inside the tower, you find a wounded guard; two others were killed by the dragon. Just as Irileth tries to question him further, the others shout that the dragon is approaching. She immediately orders the attack on the dragon.

Kill the dragon
It is the dragon Mirmulnir. The soldiers try to fight the fire-breathing monster with bows and arrows. However, the dragon is very fast, and many arrows miss their mark. Irileth attacks with a lightning spell, thereby weakening the dragon. You can support them by attacking the flying dragon with arrows or spells. Once the dragon lands, you can quickly run over and fight him with melee weapons - provided you do not get hit by his fire breath. You can also retreat into the safety of the tower and engage in ranged combat from cover.

During the fight, Mirmulnir rumbles:

"You are brave. Balaan hokoron ★. Your defeat brings me honor."

Eventually, the dragon becomes so weakened that he can no longer fly. Soon, he rears up one last time before succumbing to his injuries. As soon as the dragon lies dead on the ground and you approach him, he burns from within, and his soul is transferred to the player.

Investigate the dragon
After the dragon has burned away, only his skeleton remains. If you investigate it, you will find dragon bones and scales, gold coins, a gemstone, a few arrows, as well as the armor, helmet, and boots of a Whiterun guard.

Use your new shout
After the battle, the guards gather around Irileth and note with astonishment that the player is a Dragonborn. From ancient legends, some know that Dragonborn heroes once killed dragons and stole their powers. And what they just witnessed was nothing less than that. To be certain, you are told to attempt a dragon shout. According to the legends, only a Dragonborn can shout just like the dragons themselves without having been instructed in it.
When you use the shout, everyone is deeply impressed and compares it to the Greybeards on their mountain. Only Irileth's enthusiasm remains tempered. What she sees is a dead dragon. Thus, the realization has been gained that they can be killed. She does not need a legendary Dragonborn, she explains pragmatically; anyone who can kill a dragon is enough for her.

Report to Jarl Balgruuf
Nevertheless, Irileth admits that this was the closest fight of her life. She advises the player to return to Whiterun and report what happened to the Jarl.
Just before reaching the city, a dull thud can be heard, and the earth trembles for a moment with the word "Dovahkiin". The Jarl and his brother Hrongar interpret it as the summons of the Greybeards, and after you tell them about absorbing the dragon's soul, they are certain that the Greybeards have called you ★. They are masters of the Way of the Voice and live high up on the slopes of the Throat of the World. Since the player obviously possesses a unique talent, you are sent to the Greybeards at High Hrothgar to be instructed by them. It is an immense honor, and one must not refuse the call of the Greybeards.

Since you have rendered a great service to the city, the Jarl, by virtue of his office, appoints you Thane of Whiterun. You are assigned the huscarl Lydia and presented with a weapon from the Jarl's personal collection. With this, the quest is completed.

Tips & Notes

• The dragon flies in as soon as you approach the surviving guard in the tower - regardless of how much time you spent with the Jarl or in the city beforehand, or how far away Irileth and her companions are.
• In unfavorable situations, Irileth will face the dragon alone for an extended period. She can weaken him significantly, but without help, she usually succumbs. However, she cannot die.
• Nevertheless, it is possible to hide inside the tower and do nothing. Eventually, the dragon will be dead, and you can step outside to absorb his soul.
• Mirmulnir uses the "Fire Breath" shout at its third stage, which is YOL TOOR SHUL
• Mirmulnir's soul is automatically linked here with the dragon shout FUS from the previous quest. To equip the shout, go to the magic menu and unlock it (by mouse click on the PC). Then exit the menu to perform the shout (bound to "Y" in the original PC version).
• If you possess over 5,000 gold coins, you can now purchase a house in Whiterun from the steward Avenicci: Breezehome.
• Avenicci can also tell you quite a bit about the history of the city and Dragonsreach.