jmc

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

The distance from $(0,1)$ to $(3,4)$ is $\sqrt{(3-0{)}^2+(4-1{)}^2}=3\sqrt{2}$.

The distance from $(3,4)$ to $(4,3)$ is $\sqrt{(4-3{)}^2+(3-4{)}^2}=\sqrt{2}$.

The distance from $(4,3)$ to $(3,0)$ is $\sqrt{(3-4{)}^2+(0-3{)}^2}=\sqrt{10}$.

The distance from $(3,0)$ to $(0,1)$ is $\sqrt{(0-3{)}^2+(1-0{)}^2}=\sqrt{10}$.

Adding all of these side lengths, we find that the perimeter is $4\sqrt{2}+2\sqrt{10}$. Thus, our final answer is $4+2=\boxed{6}$.