This is a timed (30 mins.) escort mission, so do not attempt this until you are at a high enough level to easily defeat waves of Ruins 3 enemies, as well as be able to protect and heal Donoph. Robots may want to take some star atoms along, and maybe a draw weapon to deal with Gunners in the final room easily. The timer starts immediately, so head into the teleporter and meet up with him.

Open the switches leading to the next room, then hit the two on the ground, turn left to find the last two. Head right into the now open door, Beat the Claws, and head right again untill you reach a room with three doors, and a crystal in the center. To your left there are two doors, take the leftmost one. Clearing this room gives you 10 kills. Continue on, clear this room, head left and hit the switch. Head right until you get to the room full of enemies, giving you 20 and 30 kills. Continue, hit the switch in the next room, and take the door on your left. This room will give you 40 and 50 kills.

The newly opened door takes you to a room with a door behind lasers(Point A), but it's locked so return to the room with the Crystal in the center. Take the northernmost door until you reach a room full of claws. Kill them, then head north to hit 4 switches. Return to the previous room and head into the room you opened by hitting the switches. Activate the switch in this room, go back to the last room, and go through the door on the west. Activate the switches in here,(each one has a trap, careful) and go into the opened doorway. This room will give you 60 kills, and head right.

Kill the enemies in this room, and head straight to hit a switch. Now go back, and take the open door on your right(or the one heading south on your map) as a shortcut to Point A. The lasers will be down, so hit the switch to go to the final room. (The switch also opens a door on your left with a healing ring, use it if you need to.) Clear this room to get 100 kills, see the cutscene, and take the teleporter back to get your reward.

Check here for enemy counts.
Check here for quest experience counts.