"This is the Arkenstone of Thrain," said Bilbo, "the Heart of the Mountain; and it is also the heart of Thorin. He values it above a river of gold. I give it to you. It will aid you in your bargaining."
-The Hobbit, chapter XVI: A Thief in the Night
Bilbo was scared near witless of what he had just done, yet he felt he had done the right thing. Even if it cost him the entirety of his share of the gold and brought the wrath of Thorin upon his head; it had been the right thing to do, particularly if the Elf King's, Gandalf's, and Bard's reactions were anything to go on. If nothing else, it would potentially soften the edges of the confrontation he was sure was only a dawn away.
So it was that he returned to the gates and woke Bombur from his sleep. For his part, Bilbo was hard pressed to sleep now that his watch, such as it was, was done, much less listen to the thanks of a dwarf after what he had just done, the deed that they would no doubt see as a betrayal in the morning. In the end, the anxiety and excitement of the night caught up to him and Bilbo grew weary.
And so, with the thrill of the night having taken its toll, he curled up in his corner and eventually fell asleep. His deeds would soon enough come to light. Until then, he was content to dream of eggs and bacon until the morning light woke him.