King of Cards is included in the Treasure Trove update v4.0A (0004000e) for the cart dumped version of *.3ds/ dumped & converted to *.cia. If your game is the digital eShop version, you would have look for the latest dump that has the KoC campaign.
Keep in mind reinstalling a (0004000) game CIA will overwrite and erase the save. Use Checkpoint or JKSM before updating or reinstalling.
Good luck finding it.