After you give Cor Kalom the focus, he will tell you that the Sleeper has told him he needs a stronger secretion from the Minecrawlers in order to contact the sleeper. He will tell you that the answers could lie in the Old Mine. Go there now.
Apparently the minecrawler mandibles don’t contain a strong enough secretion to contact the Sleeper, so you must find the alternative. Once you get to the Old Mine, you should see some templars standing outside. One of them, Gor na Drak, will teach you how to get the mandibles from the minecrawlers. Do this, minecrawler mandibles will come in handy later.
Once you get inside the old mine, Drake, one of the Guards, will stop you and tell you about the mine. If you give him a beer, he will tell you a couple of tips about fighting the crawlers.
Anyway, head to main shaft, do all the sidequests if you haven't done them already, and then talk to Ian. He will tell you that their masher broke down and he needs a new gearwheel. He tells you that he's clueless about where to get a new one, but suggests you try a sideshaft with a broken masher.
Head over to the bottom of the mine, but detour to a small passageway with one guard standing in front of it. There should be two ladders and a giant machine with three crawlers standing around it. On the ground is the gear in question. Note that it may be difficult to exit this room using the ladders, so instead jump over to the rock entrance from the wooden platforms. Give the gear back to Ian.
At the bottom of the mine shaft, head over to the area with the massive gate and talk to the person next to the winch, Brandick. He'll open the gate for you, but only if you call a few of the Templars - Gor Na Bar, Gor Na Vid, and Gor Na Kosh - over to help defend him when the gate is open. Simply go back through the mine, find the three named Templars, and ask them to accompany you during the gate opening. When they're all gathered near the gate, ask Brandick to step aside and you can activate the winch.
Almost immediately, Minecrawlers will spawn right at the entrance of the new-opened chamber. Kill what you can - the Templars are very proficient and you may not get experience for their kills. If you have Light scrolls, use one now, because the interior of the chamber is difficult to navigate as dark as it naturally is. After killing numerous Minecrawlers while spelunking the darkened passages, you should come to an end-chamber with a rather tall, rather stationary Minecrawler - this is the queen. Kill her and grab the eggs around her at the floor.
Return to Cor Kalom and give him some of the eggs (he won't take the them all) to complete the quest. Additionally, you can go report to Raven in the Old Camp with some eggs still in your inventory and it will sound like you're being a dog-in-the-manger to the Sect Camp and not giving them any! Talk about playing both sides of the fence.