Strong Wire can be acquired with Strong Rope in the Cliffs region of the main Island.
This can be used in the following recipes:
Where to FindEditIn the Cliffs region of the Island, use 2x Strong Rope to climb up the cliffs, where you'll find a number of Cedar trees.
Hidden amongst them, barely visible, is the Strong Wire, as pictured. Enter the grove of trees from the left.
Strong Wire will re-generate every hour, so come back often!