After you watch Jona summon the Lleviathan, you can gain access to Tywll Cave to put a stop to the strange events.
Enter the Cave
On the beach outside of Tywll Cave, go to the southeast part of the beach to find a red chest. Then go west, and go past the cave entrance to find a blue chest. Then go back, and enter Tywll Cave.
Tywll Cave - L1
From the entrance, go to the right and walk north on the rocks. From the next island, go west, then south to find a chest that contains Iron Kneecaps. Then go north until you see a cave entrance on the left. Go into the cave and go north to reach a chest that contains Ultramarine Mittens. Then exit this room and go west and up the stairs. Go into the cave and open the chest to get a Mini Medal. Exit the cave and go back the way you came until you can go south on some rocks. When you reach the island, go east on the rocks, then go south and exit the cave.
Tywll Cave - Outside
Go up the stairs and go into the first cave.
Tywll Cave - L2
Go northeast to get a Seed of Agility. There's nothing interesting on the left path, so just exit the cave after that.
Tywll Cave - Outside, part 2
Go west and enter the cave.
Tywll Cave - L2, part 2
Go along the west edge of the room to reach a chest that contains a Mini Medal. Then go back the way you came. As soon as you see some rocks to the north that you can walk on, go north on them and keep going north. At the top, go east to find a chest that contains 1500 coins. Then go back the way you came, down the first set of rocks, then head east until you reach a cave entrance that has some stairs leading down. Go down them.
Tywll Cave - Outside, part 3
Head north along the rocks, then west and south to reach a chest that holds a pair of Blue Jeans. Then you can head to the east side of the room where there is a pot that you can break. Finally head north and up the stairs to Cuddiedig Cliff.
Go to the north edge of the cliff and watch the scene. Then check on Lleviathan to start the battle.
Lleviathan can only attack once per turn. He can do a normal attack that affects one person and a swipe attack that affects your whole party. He doesn't have any strengths against magic, so just use your strongest magic. Just whale on him and you'll defeat him eventually.
Afterward, you receive a Fygg. In Porth Llaffan, talk to Jona south of the church, then talk to the person dressed in blue on the boat near the item shop. Choose to sail to the next continent. Your next stop is Slurry Quay.