For one thing, the key player in this story is not the King per se, it's those stilts. And as silly as those stilts may seem, they're more than a little significant. That's because they represent the delicate balance between working hard and playing hard, each of which is equally important.
When he has the stilts to give him a much-deserved dose of fun at the end of each day, the king runs the Kingdom of Binn fairly and with gusto. But when those stilts go missing, he falls right down in the dumps because his life is no longer properly balanced. See? It's all about balance, people. Those self-help books were right (although we would like to point out that Seuss got there first).
The book ends with this line: "And when they played they really played. And when they worked they really worked" (120). Need we say more? This sums up the importance of a balanced life for both King Birtram and Eric. They know how to work when it is necessary, and they know when they've earned a little me-time, too.
This wisdom is what keeps them healthy and happy, whereas Lord Droon, who believes in working too much (and scheming when he's not working) is unhappy the entire book, never more so than at the end, when he eats his just desserts.