To complete the Grime-Encrusted Ring quest -- the objectives will not show up on your dungeon map -- place the Grime-Encrusted Ring into the The Sparklematic 5200 found inside The Clean Zone within the Gnomeregan instance. I found that only one of the three Sparklematic machines allowed me to insert the ring, so if the first one doesn't work, try the others.
Turning it in rewards a Brilliant Gold Ring . You will also receive the quest Return of the Ring to return the ring to an NPC in Ironforge or Orgimmar, depending on your faction. Bring the NPC some additional materials and he will upgrade the ring to Nogg's Gold Ring.