Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $2$ years older than Chloe, and Zoe is exactly $2$ years old today. Today is the first of the $6$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?