Note: You do not have to discover the weakness in your battle against the NM. If you end up killing the NM without intimidating it, you still get two guesses at its weakness when you report back the the NPC.
Example: No element intimidated the Mamool Ja NM. The peculiar bud scent did not intimidate him. Dropping off the ledge did not intimidate him. Eventually, he died, never getting intimidated. So I guessed that the weaknesses were: 1) a weapon skill from a certain angle, because solo, trying weapon skills from different angles was very difficult, and I only managed two different angles during the fight. 2) a specific weapon skill, because I obviously did not use every weapon skill in the game against him. I wasn't even able to use all my sword skills. I got the first weakness correct (certain angle), completing the quest, and receiving 200 Imperial Standing. I was wrong about the second weakness: I suspect that Shining Blade was not considered light element damage, or that the bud scent or dropping off the cliff didn't work out for some reason.