Irkngthand is fairly easy and straight forward all the way through. The main quest inside this dungeon is a thieves guild quest in which you acquire the Skeleton Key and end up killing Mercer, the, at that point, former master of the thieves guild.
Mercer turns invisible during the fight, but a faint shimmer in the air will give away his position. He will run quite a lot. After you have killed him the cavern fills with water. Don’t worry about trying to find an exit. You will have to wait for the cave to fill up with water and eventually an exit will open. Make sure you ascent in such a way that you will be inside the small hole of the exit.