jmc

We know that the ratio of the lengths of the sides of a 30-60-90 triangle is $1:\sqrt{3}:2$. We know that the length of the hypotenuse is $2\sqrt{6}$ and the ratio of the length shortest leg to that of the hypotenuse is $1:2$. Therefore, the length of the shorter leg is $\sqrt{6}$. Since the ratio of the shorter leg to the longer leg is $1:\sqrt{3}$, the length of the longer leg is $\sqrt{6}\cdot \sqrt{3}=3\sqrt{2}$. The sum of the lengths of these two legs is $\boxed{\sqrt{6}+3\sqrt{2}}$ centimeters.