I came here for help and felt some clarification was needed if you're in a group doing BRD anyway.
First, I obtained my 5 weapons from the digger NPC's outside the instance's entrance. Then I went inside and did a full clear. I did not use a mole machine but if you're going to solo it, that's definitely a nice way, just jump down using the Grim Guzzler option.
The Dark Anvil is what you're looking for and it is guarded by Lord Incendius and his firelord NPC's. In order to find him, you have to use the Dark Iron Highway which an area to the left when you first come inside the entrance (use the iron gate immediately to the left of where you enter, and you will see a brick road looking area, with some Dark Iron NPC's and a fire elemental or two and some Dwarven braziers). It should say "Dark Iron Highway" or "Shadowforge City" here.
To the left will be Bael'gar, a magma elemental and to the right will be an open doorway with big ugly faces that looks like the BRD load-up screen. Follow that pathway killing elementals until you reach the Lord Incendius.
Kill him. Then you just click the weapons from your bag while standing there to break them and they just go away (yay, bag space). You get credit for your quest, boom, done.
FYI: to get your bearings, beyond the Dark Anvil is the Hall of Crafting where Fineous Darkvire is found. The Hall of Crafting is a C-shaped spiral that goes to the half-circle hallway and that's connected to the vault room and the Domicile. So if you find yourself in a party that doesn't do Bael'gar, you can still hopefully navigate this way also. The Domicile is accessible through the Ring of Law, if that's the way you're headed make sure you clear all the way through down to Lord Incendius.