What does the king do to solve his problem in the story of The Adarna Bird?

The king in the story actually had two problems to solve - his sickness at the beginning of the story and the conflicting claims of two princesses who wanted to marry his youngest son Don Juan at the end of the story.

In the Filipino epic, Ibong Adarna, the king loved his youngest son Don Juan so much that that when one night, he dreamed that two traitors conspired against Don Juan, the king became so frightened and depressed that he did not even want to eat or take a rest, and fell sick with a malady, of which none of the physicians of the kingdom were able to cure. The king was advised that the Adarna bird was the only creature in the world which could restore to him his lost health and tranquility by using its song.

Acting on this advice, he sent out his oldest son Don Pedro to look for this coveted animal. Unfortunately Don Pedro fell asleep beneath the tree where the Adarna was accustomed to pass the night and the bird defecated on him, turning him to stone. The second son, Don Diego, met the same fate. The youngest son Don Juan succeeded where his brothers failed and not only captured the bird, but rescued his brothers, turning them back from stone to living humans.

Even though he had rescued them, his brothers were jealous and conspired to beat him, leaving him unconscious in the middle of the road, and take the bird back themselves. Don Juan has several more adventures climaxing in him returning to the king with the princess Doña Maria in tow whose hand he had won through his great deeds. Waiting from Don Juan at the palace was another princess, Doña Leonora, who also wanted to marry Don Juan. The king consulted with the archbishop of the kingdom on the case and the church dignitary decided in favor of Doña Leonora. Doña Maria was not willing to give up Don Juan so easily however and used magic to bring on a great flood that threatened the whole kingdom, so the king convinced Princess Doña Leonora to be content with marrying Don Pedro, the brother of Don Juan, which she did for the good of all and Don Juan married Doña Maria.