Bilbo only leaves Thorin's side -- once his own wounds are covered for the hobbit had done more scouting and hiding than actual fighting no matter how good with Sting he had gotten -- to run reconnaissance on his nephews and the other wounded. Oin is with them both and they hold hands when awake across the scant distance between their beds.

Bilbo relates ll of this in detail, sparing Thorin nothing.

He comes back that evening looking pale in the firelight of the tent Thorin is in, poultices newly changed, bandages and wounds washed, and falls to his knees in tears.

"They've found a plant that will cure Kili or Fili of their fever, and save his life," Bilbo weeps, though there is no such plant, simply a small he has learned from Smaug and the Ring...one he can perform as many times as he'd like, though he'll not tell Thorin this. "We have only enough for one. Oin asks you to choose."