Start by talking to Larrissa just outside of the lighthouse. She will explain her boyfriend disappeared, leaving the lighthouse unlit, and locked the front door. Agree to help. Now head east until you reach the broken bridge.
Use a plank on one side of the bridge, then cross the bridge and use a plank on the other side. Now head back across and cross the basalt rocks to the south of the lighthouse. Now enter the barbarian agility course at the outpost.
Look for Gunnjorn walking around the course. Talk to him about Horror from the Deep. Get the key from him and return to the lighthouse. Now unlock the door and enter the lighthouse. Larrissa will follow you through.
Talk to her and she'll ask you to fix the light. Climb up the staircase to the top floor. Use your molten glass on the lighting mechanism to repair the lens. Now use your swamp tar on it to make the torch flammable again.
Finally, use your tinderbox on the mechanism to get it working again.
Go down and talk to Larrissa again. Climb down the iron ladder and study the strange wall. Close out and use your elemental runes, arrow, and sword on the strange wall.
It is recommended to now leave and get all of your combat gear. You will need runes for all of the elemental spells, arrows and ranged equipment, and melee. The different attack styles aren't necessarily required, but will be extremely helpful.
When ready, enter through the eastern door next to the wall and climb down the ladder. Jossik will be nearby, injured. He will say that the creatures will be back again soon.
A level 100 Dagannoth will come out of the water and start attacking you. It only uses melee, so shouldn't be too hard to kill with protect from melee. After killing it, talk to Jossik (if it says you are too busy to talk to him, climb up the ladder then go back down).
Afterwards, the Dagannoth mother will appear. She changes color, and each color represents either an elemental magic spell, melee, or ranged. She can only be damaged by whatever the color she is represents.
She will change colors randomly, and uses ranged and melee. If you protect from one style, she will use the other. Also, you cannot protect ranged and hide, because she will disappear and you will have to start the fight over.
After killing her, quest complete!