Two-Digit Numbers with a Digit-Sum Condition

Let $b \geq 2$ be an integer. Call a positive integer $n$ $b$-eautiful if it has exactly two digits when expressed in base $b$, and these two digits sum to $\sqrt{n}$. For example, $81$ is $13$-eautiful because $81=6$$3_{13}$ and $6+3=\sqrt{81}$. Find the least integer $b\geq 2$ for which there are more than ten $b$-eautiful integers.