As a lowly page, Alanna doesn't see the king of Tortall up close and personal very often. She does know that he's usually smiling, though, which makes it a big deal when he enters Jonathan's chamber during the Sweating Fever, "looking worried. Deep lines seemed permanently carved around his mouth" (4.112). Makes sense; you'd be freaking out too, if your only son lay on his deathbed!
He takes the time to talk to Alanna, however, and expresses regret that he has not stepped up and taught noble boys with the Gift more about magic—it might help in times like this. The fact that Roald seems willing to evaluate his past decisions and change his mind makes us think that he's probably a good person to be king and helps explain why his son Jonathan seems like such an all-around good guy.