Who becomes king under the Mountain after Thorin's death?

Dáin Ironfoot becomes King under the Mountain after Thorin's death. Dáin is Thorin's second cousin and the chief of the Dwarves of the Iron Hills. He arrives with a large force to support Thorin during the crisis at Erebor.

Before the Battle of Five Armies, Thorin sends for Dáin because the Men of Lake-town and the Wood-elves are demanding a share of Smaug's treasure. The dispute is interrupted by an attack from Goblins and Wargs, forcing the dwarves, Men, and Elves to fight together. Thorin is mortally wounded in the battle and dies soon afterward.

With Thorin and his heirs Fíli and Kíli dead, the kingship passes to Dáin. He becomes Dáin II Ironfoot and rules the restored Kingdom under the Mountain. Under his reign, Erebor grows prosperous again, and relations with Dale and the surrounding peoples improve.

Balin is an important member of Thorin's company but is not his successor. Fíli is Thorin's nephew and heir, yet he dies before Thorin. Thorin himself cannot remain king after his death, making Dáin the canonical answer.