jmc

We look at the lateral area of the cylinder as a rectangle (imagine a peeling the label off of a soup can and laying it flat). The length of the rectangle is the circumference of the base, $12$ inches in this case, and the width of the rectangle is the height of the cylinder, $5$ inches. The spiral strip goes from one corner of the rectangular lateral area to the other, so it is also the hypotenuse of a right triangle. We find the length of the hypotenuse with the Pythagorean Theorem, or we recognize that $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$, so the length of the hypotenuse (the spiral strip) is $\boxed{13}$ inches.