jmc

We use the distance formula to find the length of each side.

The distance from $(0,1)$ to $(2,5)$ is $\sqrt{(2-0{)}^2+(5-1{)}^2}=2\sqrt{5}$.

The distance from $(2,5)$ to $(5,2)$ is $\sqrt{(5-2{)}^2+(2-5{)}^2}=3\sqrt{2}$.

The distance from $(5,2)$ to $(7,0)$ is $\sqrt{(7-5{)}^2+(0-2{)}^2}=2\sqrt{2}$.

The distance from $(7,0)$ to $(0,1)$ is $\sqrt{(0-7{)}^2+(1-0{)}^2}=5\sqrt{2}$.

Adding all of these side lengths, we find that the perimeter is $10\sqrt{2}+2\sqrt{5}$. Thus, our final answer is $10+2=\boxed{12}$.