Would that mean his realm or the shivering isles no longer exists?
Well first off, when you beat the shivering isles quest you become the ruler of the shivering isles so no. Plus before that quest started it was stated that sheogorath and jygalag were essentialy the same person but different personalities. Sheogorath was the one who built the kingdom then jygalag invades his kingdom to restore order then sheogorath rebuilds his kingdom again. (This all goes on and sheogorath and jygalag are the same person.) Hope i didn't spoil anything for you o.o