Heritage o' the Dark Iron

If anyone else is a bit confused as to how to actually get there (I went to the dungeon, thinking that was it):
Use your mole machine.

Don't know why I didn't think of that at first, but that's what got me there in the end :)

This quest is confusing as you do not need to head into the dungeon. All Dark Iron Dwarfs have a racial spell called Mole Machine. It is located in your spellbook under the General Tab. You will need to use this outside anywhere in the game and select Eastern Kingdom > Shadowforge City. This will bring you into the Blackrocks Depth but everything is friendly now. Seems that this place has a good and bad side. The NPC Moira Thaurissan will be located in the middle of the lava pit beside the Anvil. You can cash in the quest. You will only get the Transmog set and not the items in your inventory. Hope this helps.

If for some reason your mole machine is on cooldown, there is a spare in Stormwind in front of the embassy that goes only to Shadowforge City.