3.113: Piecewise velocity; find $k$ from distance

A particle travels in a straight line so that, $t$ seconds after passing a fixed point, its velocity, $v\,\mathrm{ms}^{-1}$, is given by
$$v = \mathrm{e}^{\frac{t}{4}}\quad(0 \leqslant t \leqslant 4),\qquad v = \dfrac{16\mathrm{e}}{t^{2}}\quad(4 \leqslant t \leqslant k).$$
The total distance travelled by the particle between $t = 0$ and $t = k$ is $13.4$ metres. Find the value of $k$.