There's simply no way to talk about The King's Stilts without mentioning those red stilts straight away. After all, they appear in the title of the book. The stilts in The King's Stilts represent the balancing act that the king must maintain in his professional and personal life. It doesn't hurt that they are actual objects that people balance on, either. Seuss did like to hammer a point or two home.
When the king's stilts (or work/life balance) are lost, he becomes a mopey mess. He is unhappy and doesn't perform as well as he usually does. The downstream effects of this include the loss of balance between the Kingdom of Binn and its natural setting. It is only when the stilts are restored to King Birtram that he's able to find balance for himself… and for his kingdom. We think that says it all.