Just a piece of paper

Referring to the diagram, let the length of the diagonal segment of positive slope be $x$ .

Now the base side is divided into two segments: the one to the left has length

$a = \sqrt{11^{2} - 8.5^{2}}$ , and hence the one to the right has length $11 - a$ .

Looking now at the small triangle on the lower right of the diagram, we have that

$x^{2} = (8.5 - x)^{2} + (11 - a)^{2} \Longrightarrow x = \dfrac{8.5^{2} + (11 - a)^{2}}{17}$ .

The fold then has length $\sqrt{11^{2} + x^{2}}$ , which when we plug all the numbers in to a handy calculator gives us a fold length of $\boxed{12.167}$ to $3$ decimal places.

Let A be the point on the bottom left corner, B be the point on the top left corner, C the point created by the fold, D the point on the bottom right corner, and E the point on the top right corner.

We know the segment AB has a a length of 8.5 and BC has a length of 11. Using Pythagorean's Theorem, we can solve for the length of AC, which is sqrt{48.75} . We also know the length of AD is 11 and AC+CD=AD. Solving for CD, we get the length to be 11-sqrt{48.75}.

Next, we know the following angles are complimentary ACB and ECD, ABC and ACB. If two angles are complimentary to the same angle, then they are congruent. Therefore, ABC equals ECD. Now we can conclude that triangles ABC and CDE are similar and have proportional sides.

We can set up the following: BC/AB=EC/CD. Solving for the side EC, we get its length to be approximately 5.1996.

Finally, using Pythagorean's Theorem on the triangle BCE, we can solve for EB which is the the length of the fold, having a value of approximately 12.167.