The complex number $z$ satisfies the equation

$$
z^{2}(2+\mathrm{i})-8 \mathrm{i} z+t=0
$$

where $t$ is a real number. It is given that one root is of the form $k+k \mathrm{i}$, where $k$ is real and non-zero.

Find $t$ and $k$, and the other root of the equation.

[8]