This trophy requires that you complete each request on Eternal mode, which is the same as Katamari Forever mode but without a timer.
Eternal mode usually became available for me after completing requests on Katamari Forever and Katamari Drive modes. If it wasn't available, I'd replay one (or possibly both) and try to score 100, which then would unlock Eternal mode.
Only four of the King's requests have Eternal mode:
Wake up the King
The King's Dream
Only three of the Roboking's requests DO NOT have Eternal mode: