Answer: 32

Small Hint:

Place the unit square on a coordinate plane and write equations for two parallel construction lines

Big Hint:

The distance between the parallel lines is the side length of the small square

Solution:

Put the outer square at $(0,0),$ $(1,0),$ $(1,1),$ $(0,1).$ One pair of construction lines has equations $$\begin{aligned} nx-(n-1)y&=0,\\ nx-(n-1)y&=1. \end{aligned}$$ The other pair is perpendicular to this pair, and the two pairs have the same separation. Thus the inner square has side length $$\frac1{\sqrt{n^2+(n-1)^2}}$$ and area $\frac{1}{n^2+(n-1)^2}.$ Hence $$n^2+(n-1)^2=1985,$$ or $n^2-n-992=0.$ Its positive root is $n=\frac{1+63}{2}=32.$