How do I recover a deleted 'Royal Kingdom' game on my iPhone?

You can redownload the game:

Redownload apps and games from Apple - Apple Support

If you are looking to recover game save data, then that would depend on where the developer of the game saves it. If it was saved on your device then the game data has been deleted with the game. There are other options that the developer can use to save game data, such as iCloud or on their own servers where the data would not be deleted, but not sure where it is saved for that specific game.