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Kili is all right the moment he sees that Fili wasn't crushed by the storm giants. He breaths a sigh of relief, and then in the following commotion of saving their burglar, he forgets that Fili was almost lost. He takes his place at Fili's side and despite the chill from the rain and stone floor, he is asleep in moments.
It is Thorin who stays awake. He lies with his back facing the mouth of the cave, something he would never do if his mind were properly working, but can't keep his heir's frightened face from his mind, the shrill cry for his brother that rang into the night when they were separated. His back is the cave's mouth so he can watch over Fili.
He is not an affectionate dwarf, cannot bring himself to do as he wishes and wrap his body around Fili's, and Kili's as well, to hold them to his chest and feel them breathing against him to remind himself that they are ok, that he hasn't failed his sister and let them fall. But he is a king, even if he is without his mountain, he stays were he lies and watches his nephews, obsessively counting each rise and fall of his chest, closing out all other sounds in the cave but Fili's breath and quiet sleep-sounds, and does not sleep.