Beitild owns the Iron-Breaker Mine in Dawnstar and she is nearly always near the guards. Since there are no special conditions for the quest, any method that ends with her dead will count as a success. There are several ways to go about this:
- Claim to be from the Dark Brotherhood. She will draw her pickaxe. At this point, run out of town. When she follows, kill her.
- Wait until night, then break into her house and kill her there, where there are no witnesses. This will be especially easy if the Dragonborn has a high Sneak skill and is able to attack her while she is asleep.
- With high Sneak, jump onto the roof of a building and use a bow to snipe her in one hit without the guards seeing it. (Poisoning the bow will help.)
- The mountainside above the mine (near Nightcaller Temple) provides a safe, albeit distant, vantage point from which she can be sniped. Enabling quest markers may make it easier to distinguish her from the other NPCs around her.
- Reverse-Pickpocket poison into her inventory (with the "Poisoner" perk).
- Sneak and cast Fury, Frenzy, or Mayhem, and the guards will attack her. The perk Quiet Casting helps here.
- Being in Werewolf form will allow the player to avoid a bounty on their head, no matter how overt they are. As long as no NPCs see the Dragonborn change, they will not know who it is.
- If a dragon shows up, it may kill her, but with so many other NPCs around it might be a challenge to keep the fight going long enough to do so, and other NPCs (such as Rustleif, the town's blacksmith) may also die in the process.
- Wait until night and break into her house. Claim to be from the Dark Brotherhood, and she will draw her pickaxe. Do not attack, instead, exit the house, where the guards will take care of her.
- Cast a flaming familiar near her, where she will be killed by the blast when its timer runs out. Be sure to stay near Beitild so the familiar is right next to her.
When she's dead, return to the Sanctuary and collect the reward from Nazir.
I represent the Dark Brotherhood. Your death is at hand! "What? Well, we'll see about that now, won't we?"
Someone wants you dead, Beitild. "And let me guess, that someone is my so-called husband Leigelf? Well the feeling's mutual."
(Remain silent) "I've a mine to run, so if you'll excuse me..."
The first option will turn her hostile, the second and the third will make her walk away.
I must go to Dawnstar and kill the mine boss, Beitild. I may use any means at my disposal. When I am done, I am to report back to Nazir at the Sanctuary and collect my reward.
Betild [sic] is dead, and I have collected my reward from Nazir at the Sanctuary.