Solution #3

Because AEB and DFC are both 5-12-13 triangles, they are both right. Now add corresponding triangles on sides AD and BC:

The resulting quadrilateral is a rectangle, because all of its angles are right. And it is a square, because each of its sides is composed of two right triangle legs, one of length 5 and one of length 12.

Thus it is a square of side length 17. EF is a diagonal of that square, and hence has a length of 17 √ 2