The King of all Cosmos appears at the end of the Bird & Elephant stage. You must be really big to roll him up- over 3000m! You'll almost certainly have to bounce him first, then roll him up when he is dazed. Here is a video where I roll up both the King and Queen in the As Large as Possible version of the level:
The King will appear on the level when you reach about 1200m.
This is a very long level so you need to be sure that you are using your time efficiently to roll up big things and grow as fast as you can. Time is at a premium.
Once you get to about 1000m, be sure to roll up islands, land chunks, and especially big clouds as they add the most size to your katamari.
After you bounce the King, be sure to roll him up ASAP or he may wake up and scoot away permanently.
Oh, and per Jamone the Afro, this can be done in Eternal mode, which is unlocked after you complete the level on both As Large as Possible and As Fast as Possible.