Defeat the Gorishi Hive Queen once she is summoned.
Bring the Gorishi Queen's Brain to Alchemist Pestlezugg in Gadgetzan.
Here's the lure. Inside the hive network should be a crystal of some sort. We think the silithid use the crystal as some sort of hive-wide attunement device. As for where this crystal might be, I'd say try the hatcheries inside the hive.
Using the lure on the crystal should roust out the queen, but be warned - she will probably be protected by several guards. Take out the queen and remove her brain. As icky as that sounds, we desperately need to study it. Good luck, <name>!
Fantastic, <name>! We've won quite the victory here today. It will take some time for us to fully understand what we have here; we'll put it to good use though, don't you worry about that.
As for the silithid, I genuinely doubt that this was the full extent of their presence. This is much too clean of a resolution, I'm sorry to say.
Upon completion of this quest you will gain: