The Island Heart can be obtained by completing the purple quest on the island.
To unlock the purple quest, you have to complete 5 yellow quests over the time span of 5 days.
A new quest appears each day. To complete each one, you must transform into a Dooky by buying a coin from the merchant. You can trade for the coins by exchanging the trash you get from the island.
For one of the quests, you need a Dooky Calibar, the heroic sword trash item you get from Dooky Island.
