This is an annoying quest, so hopefully this will make it less annoying for other people. Be mindful that you need to be able to kill the monsters relatively quickly in order complete this quest on very hard+.
Ideal path is to take the data from monument 1 first on east and south (the right teleporter is the third one you see when entering the room).
Then head on north to the purple switch to clear the path to collect data from monument 2. Then go and collect it (ignore the switches to the boxes if you have trouble clearing monsters fast).
Then proceed to the monument 3 and then get ready to teleport to monument 4.
It is important that you have collected the data from all other monuments, because only if you had done so an earthquake will happen and a teleporter towards the monument 5 will appear. Be prepared for the alien-demon bloodshed (and be sure to kill them fast).