Why is the minimum runway length non-linear in relation to the landing weight?
While the kinetic energy is proportional to mass, it is not proportional to velocity. 

The heavier the aircraft is, the higher the landing speed. So the stopping distance increases mainly because of the higher landing speed.

Comparing two landings, first at mass = $m_1$, the second at mass = $m_2 =1.1 \cdot m_1$, the landing speed will be $\sqrt{\frac{m_2}{m_1}}$ higher during the second landing.

The kinetic energy during the landing will scale proportional to the square of the mass.