If you are level 70 you can do this solo even without stealth. Helps to be a mage to blink past a few ""tight spots"".

Enter BRD, go through the locked door on your left, continue straight to the ""highway"", take a right and follow the hall to Incendius, melt his face, loot corpse, dance happy jig, disenchant BOP drop (if you can), continue straight and go up the ramp, avoid mobs by walking on the railing most fo the way. At the top fo the ramp go through the doorway and go to the left, take the next left into the Cofffer Room. Blink past the boss and the dog so you are standing against the Dark Coffer. Attack and kill the dog and ignore the boss until the dog is dead. At level 70 you should resist all or most of the bosses spells. Kill the boss, loot and dance again, DE, walk to the edge of the room (stand about where the boss was standing), adjust your camera angle so you can click the picture on the wall and read the caption n the picture to find where the NPC is that you need to kill to get the Dark Coffer Key. Backtrack to wherever that mobs is and kill him to get the key, then return to the Coffer Room, open the Dark Coffer, loot contents, hearth out to Orgrimmar or Ironforge... spam the trade channel til you sell the ""enchant"" or get ignored by everyone on your server.

If you happen to get ignored by everyone on the server, pay Blizzard to do a server transfer to another. Repeat as needed.

:-P

with thanks to Zaber's comments I just did this a couple of times to help out the tanks for the illidan encounter. I would like to add the following:
- you can have multiple Burning Essence in your bag so you can reset the instance over and over (well untill 5x max per hour) until you have all the needed Burning Essence.
- you can have multiple Lesser Arcanum of Resilience in your bags aswell (they don't stack though)

I just sent all my Libram's through the mail and went flying back and forth to Redridge to pick them up.

This information below from another site (http://wow.allakhazam.com/db/item.html?witem=11751;page=1;howmany=50#1165864703213263922 source) helped me out aswell.
The list of Dark Key holders and where they can be found are:
Dark Keeper Offgut West Garrison - room with Angerforge
Dark Keeper Bethek Patrols near Coffer room
Dark Keeper Vorfalk Grim Guzzler Tavern
Dark Keeper Zimrel Arena; upper seating area
Dark Keeper Uggel Top of ramp in Golem Room w/Fineous
Dark Keeper Pelver Domicile room (storeroom in center)

All but 2 are fairly easy to do.
Bethek spawns almost instantly outside the Coffer room, so be prepared.
Vorfalk is surrounded by patrons in the Grim Guzzler. To get to him, you have to pull the patrons around him, 1 by 1, with the Dark Iron Ale.
Just make sure you kill the patron before they drink it, as it's limited in supply.
After all that, the Burning Essence is still a 50% drop, so you may need a few attempts before you get it.