@Winchan I got an impression the duke's family is an economic powerhouse. However, the royal house should be in command of the armed forces. Possibly the duke has some forces of his own, but I doubt it would be enough considering the duke's apparent jolly and relaxed personality. Many other noble families would rather side with the king in a sudden, uncertain situation. The nobles now seemingly supporting the Albert family might try to instead get their share of the loot when the dust settles. After all, going against a ducal family would be far more safe than going after the royals. It would take a whole lot of prepation to pull off a coup d'etat successfully.
Although the royals might consider it strange, but I don't see them rejecting a proposition for the Albert house to voluntarily spend much of their wealth and thus much of their influence in some grand national project. They could also spin off a part of their family, thus splitting their power in two. I bet the king, assuming he's not too much of a tyrant, would prefer that to actually crushing an important family. Internal disputes in a country are rarely good for business and foreign relations.