What I'm saying is that IF a Daedric Prince was defeated while inside their realm, would they just be reborn, or would they be destroyed, and their realm with it.
They are the realm, the realm is them. That silly Scottish man on the throne of madness is nothing more than an avatar for mortals to comprehend. They have all the power in their realm, for they are the realm. If another prince went to another prince's realm or Mundus, it has to be in the form of an avatar, because they cannot physically be there as they are the realm.
http://www.imperial-library.info/content/morrowind-monomyth It has some bits how they formed their realms.
If that's the case, wouldn't the Shivering Isles be dead?
You're just killing an avatar.
Sheogorath left, the Champion became regent,
No, the champion became the realm as the new Sheogorath through some mythic mumbo jumbo you need to bush up on.
then Jyggalag was defeated, so the essence of the Isles would be destroyed, right?
No, as I said earlier he is no longer Sheogorath, the Champion of Cyrodiil is Sheogorath. Jyggy is now Jyggy.