If you receive the message "You find a chunk of mine gravel, but your pickaxe breaks in the process," you do successfully get the mine gravel.
At (I-8) and take the elevator to the third floor. At (I-6) to find a refiner. Trade Mine Gravel to Refiner Lid. Pull the lever on the refiner to activate it. Drop off ledge to floor below you. Pull second lever.Mythril Sand will be automatically put into your inventory.
If she gives you Trust: Kupipi, you have to zone out and back in to trigger the 2-3 mission cutscene. If you talk to her without zoning, she just talks about Nanaa Mihgo instead.
Everyone in your party must have this item to enter the BC, or have completed 2-3 previously in their current nation of allegiance. One person cannot get others into the fight who do not have the Dark Key, even if they are currently on this part of the mission, unless they have completed the mission in their current nation of allegiance.
If you are solo, make sure you are a minimum of level 30 and on a very strong job, such as a Monk using Hundred Fists. Kill the dragon before killing the Searcher, or the dragon will kill you rather easily after the Searcher has weakened you. The Searcher does not melee you unless you are close to it or it has less than 50% of its HP, so it is relatively inactive if you focus on the dragon first.
Once the Black Dragon is defeated, another cutscene will happen.
A glitch may prevent this cutscene from occurring. If this happens, you can view the cutscene here:  (begin at 8:53). Alternatively, you can exit the dais and reenter from the main entrance to watch from the Goblin Footprint.
If you are solo, make sure you are a minimum of level 30 and on a very strong job, such as a Monk using Hundred Fists. A solo hero with 4 trusts should have little difficulty clearing this BCNM below level 30.
A glitch can prevent the victory cutscene from occuring. If that happens, you can watch the cutscene here:  (begin watching from 1:53), or you can exit the shrine and reenter from the main entrance to watch from the Goblin Footprint.
After you've won the BCNM you can exit back into Palborough Mines travel a short distance on that map to H-8 where you can board one of the lever activated ferries and return to Zeruhn Mines which is connected to Bastok Mines.