Tri-circle

Let $v = AI$ , $w = GH$ , $x = FG$ , $y = EF$ , and $z = DE$ .

Since one side of the equilateral triangle is $2 + 13 + 1 = 16$ , $v + 7 + w = 16$ and $x + y + z = 16$ .

According to the intersecting secant theorem : $v(v + 7) = 2(2 + 13)$ , which means $v = 3$ , and since $v + 7 + w = 16$ , $w = 6$ .

Also according to the intersecting secant theorem, $x(x + y) = 6(6 + 7)$ and $z(z + y) = 1(1 + 13)$ . Using these equations with $x + y + z = 16$ and solving the system of equations gives $x = 10 - \sqrt{22}$ , $y = 2\sqrt{22}$ , and $z = 6 - \sqrt{22}$ .

Therefore, $EF = y = 2\sqrt{22} \approx \boxed{9.38}$ .