You have to kill monsters in three rooms to get the data to finish the quest. The only tricky thing is that you have to protect Elenore, and that the waves of enemies don't appear until you reach the center of the room. If you're having trouble, try escaping to a doorway and using long range weapons/spells.

Go down to Ruins 1, hit the switches in the first room to open the door to the first test area. Three waves of enemies will appear, beat them, and get the data. Turn right, and take the leftmost door of the two doors. Head straight, activate the switch in the next room, and clear the monsters in the following room. Take the warp, open the door, and hit the three switches to open the next doorway. Kill the one Belra and head left to reach the next test area. Three waves of enemies appear, simply kill them and get the data.

The door heading east has a healing ring, but if you don't need it, simply head south to the next room. Head to the door on your left, for the last testing area. This last area will have 6 waves of enemies including Belra, Delsabers,and Chaos Sorcerors, so be more careful. Clear the room to get the data, take the pipe up and talk to Montague to finish the quest.

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