Gandalf sympathized with Thorin's despair at the loss of Kili. Any lost soul, Kili's included, was in danger of becoming enslaved to Sauron if trapped on Middle-Earth. But in the grand scheme of the battle between good and evil, Kili was sadly inconsequential. He was no great necromancer. He held no secret maps or keys. He was one lone young person who had died tragically in battle. Kili's soul would not tip power in Sauron's favor. Searching for him would surely distract Gandalf, when the search for the One Ring had to continue.
Thorin need not guess the reasons for Gandalf's hesitancy.
"We are not friends." Thorin told Gandalf bluntly, "Wizards are not friends with the line of Durin. You do not want to help me this time because our plans do not coincide."
Gandalf stiffened at the accusation, but remained silent.
"You only helped with the my quest to reclaim Erebor because the destruction of Smaug served your purpose. You did not care if every Dwarf died in the process. We were tools to you - not friends, not brothers in arms. You think I did not know that?"
"And how was your part of the bargain different?" Gandalf asked.
"I was no different." Thorin agreed, "I was ready to make sacrifices - of lives that were not mine to give - for what I thought was the greater good. Many more Dwarves will be born, and thrive, because my people have a homeland again. That gladdens me. And you have your strong allies in the North when the time comes. Dain Ironfoot and Bard the Bowman are good men, and they will see their Kingdoms rebuilt to hold the tide of Sauron. Many good men would look upon what we did and say our results justified our methods.
I say that we alone can judge if we are good men, and the tipping of the scales will not be by one great battle or deed, but by many small ones. How can we call ourselves good men if we make such judgments about what rises high enough for us to take action? To say Kili was not worth the effort brings you low. Bilbo taught me that - the importance of the small battles, the small choices. I thought you already knew that because of your friendship with him.."
"You stopped translating, Gandalf." Bilbo noted. "Is there something wrong? Thorin is not trying to convince you, is he? Because it should not take that much time."
Gandalf looked into Bilbo's eye. Bilbo had complete faith that Gandalf would help. Gandalf could see that Thorin was right. To disappoint Bilbo, in what was so obviously a cause for good, would be the start of much greater losses in Gandalf.
"Of course I will help, Bilbo, "Gandalf assured him. "We are just at a loss of where to start. I do not even know this girl Thorin speaks of. She may not be alive. The Elves of Mirkwood suffered terrible losses and not just in battle - their settlements were attacked by Sauron's surrogates."
Bilbo eagerly provided information "Rhavaniel was at the Battle of Five Armies. I know she survived. I remember her helping the Dwarves clear the dead from the battlefield. I don't know what became of her after that. Balin said goodbye to her, and watched her run after Thranduil's troops."
"Then we must go back to the Woodland Realm."
"I know where she lived and worked." Bilbo said, "I know the names of her entire family, what she likes to eat for breakfast, her favorite song. I can lead you straight to her."
Thorin was gladdened by the news, "I have been to Mirkwood and did not see her, but perhaps I did not know where to look and only missed her. I need Bilbo's help as well, Gandalf."
Gandalf turned to Bilbo, "Do you want to come with us, Bilbo? This is a great deal to ask."
"Of course I will go. I can't imagine anything more important."
Gandalf chuckled. "It will not be like our first quest. We will have good weather, safe roads, no need for secrecy. We may be in Mirkwood this time next month."