Estimating Plane Pose Without Knowledge of Intrinsic Camera Parameters
I think there is a mistake in those calculations. 


How can you assume that $K$ has only a scaling effect? The principal point is integrated in $K$, it's not just a matrix of focal lengths.
How come is this true?


$$det(K)K^{-T} = \left[\begin{matrix} 1&amp;0&amp;0\\0&amp;1&amp;0\\-u_x&amp;-u_y&amp;1 \end{matrix}\right]$$

You don't end up with such equal numbers in the diagonal if you compute $det(K)K^{-T}$. 

To me, your question is a bit unclear and I guess some assumptions are fishy.