Well, when you get tackled by a Toaster without putting up a bit of a fight, it makes you look weak.
I think it's safe to assume Jenka in her present state is weak, weak, weak!
She states it plainly, "Even a witch like me grows old and frail..." or something.
As for the Core, I think its ... a biological living seed thing with immense power. Jenka setup the labyrinth in order to protect it. I think it must be exactly what all the characters state it is. It makes the island float. EVEN Misery says it. Personally I think it has been there even well before Jenka's time. The Core is made up of the MAIN seed, with plenty of immature seeds floating around it.
Perhaps Ballos is trying to make the island fall, maybe either by controlling the seeds directly or by applying massive force on the island...
But as long as the main seed is up it can resist Ballos. But once the main seed is destroyed it begins to fall... Until Ballos is defeated. Which then the immature seeds have enough strength to make it float on their own [without the force of Ballos on it] An idea anyways...
Another thing to think about is, Kazuma says another way to turn the enraged Mimigas back to normal would be killing the Core. That must mean the Core must have some kind of connection to magic and the Red Flowers. Maybe the flowers connect you to the power of the Core by magic or something....
Oh and what about the Undead Core? Well, applying my theory that Igor had been feeding the Sky Dragon Eggs red flowers stuff (Except for egg00) which when hatched they ended up as Zombie Dragons. Well, the Doctor probably used the Red Flower / Crystal to do the same to the Core. Turning it into the Undead Core...
If you agree on that theory that means the Red Flowers are like super magic steroids in someways. They make Mimigas insanely big and strong, they can turn the Doctor into the Muscle Doctor... They make the Sky Dragons into Zombie Dragons... They make the Core on the verge of death into the Undead Core...
They can made the dead alive or increase the living's power and size...
Even when the Doctor died or should have died, he continued to live thanks to the concentrated power of the Red Flower (the Red Crystal). But, when it was destroyed the Flower's power gave way, the island started to fall...
Also, maybe the Core safely feeds on Ballos' power, acting as a kind of natural defense. Either that or more likely Ballos' influence or magic affects the island somewhat.
Anyways ...