King's Rod is sold beneath the Statue of Sovereignty near Keeper's Altar. Budget for the currently documented 100,000 C$ rod price and the elevator trip, then use the skeleton beside the ledge as the final landmark.
Budget for the Rod and Elevator

Carry enough C$ for both the rod and normal elevator access. If you already own permanent elevator access, the travel cost may not apply. Read the live prompts rather than handing over all available currency based on an old total.
Take the Statue of Sovereignty Elevator

Enter the mine area at the Statue of Sovereignty, find Cole, and use the elevator toward Keeper's Altar. If you are trying to enchant a rod instead, continue to the altar; King's Rod is found on a nearby side path.
Find the Pedestal Beside the Skeleton

After leaving the elevator, turn toward the small ledge and look for the skeleton landmark. The rod sits beside it. Confirm that the item name is King's Rod, since similarly named late-game rods are different items with different routes.
Decide Whether Steady or King's Comes First
King's Rod is a larger general-purpose purchase with strong weight utility. If 100,000 C$ is still distant and difficult fish are the immediate problem, Steady Rod may be the more useful bridge.
Fix a Wrong Turn
If you reached the altar but cannot see the rod, return to the elevator exit and search the ledge and skeleton side rather than the altar platform. Recheck the live location after a major Statue of Sovereignty update because the underground area can change.