jmc

The graphs of $f(x),g(x),h(x)$ are all lines, and we have a segment of each, so we can extend these segments to form the superimposed graphs of $f(x),$ $g(x),$ and $h(x)$ on one set of axes:



The graph of $k(x)$ consists of the "bottom surface" of this tangle of lines, shown here in light blue:



Both pieces of the graph of $y=k(x)$ have slope $\pm 2$, so the total length of this graph along the interval $-3.5\le x\le 3.5$ is $\sqrt{7^2+(2\cdot 7{)}^2}=\sqrt{245}$. Therefore, ${\ell }^2=\boxed{245}$.