When LKHAN is destroyed, a "Sithis-shaped hole" is left, the Serpent constellation, correct? SHEOG is described as a Sithis-shaped hole. When the Exactness of Mundus is destroyed, Jyggalag himself is "destroyed" in the process, flipping him into Sheogorath, or Madness, or perhaps more specifically, Randomness.
This is a point of debate. There are numerous theories, all being based on if you believe Malacath is or isn't Triniamc, but rather his "other"
When Mundus was created, so were mortals, thus, sow as free will. The entire Arubis was then polluted with free will, which was new. Before that, all the gods were locked into actions based on their spheres. They were all predictable to a mark, and Jyg used this extreme weakness all the gods had to gain power (as we was the master of predictability). Thus, in a way, Jyg reflected the very Arubis itself. This is where things branch.
Some believe that Jyg's "other", Sheogorath, had always existed but was weak beacuse the Arubis favored Jyg (as it was still very predictable). However, after Shezarr created free will, the Arubis was polluted with it, thus letting out Sehoograth and repressing Jyg. It is through this theory that Jyg's return implies an end to free will and thus the looming destruction of Mundus.
The other theory simply states that as Jyg watched the Mundus come into creation he saw free will enter the equation. Knowing that he could no longer keep things in predictable order, he went insane and his sphere collapsed into he sphere of madness.
Some other believe that once Shezarr "died" his sphere, freedom, had to find a new avatar and latched itself onto Jyg, who then became Sheogorath to take over his new sphere.