You were at: $\frac{∂^2z}{∂t^2} = \frac{∂}{∂t}\left(\frac{∂z}{∂x}\right)\frac{∂x}{∂t} + \frac{∂}{∂t}\left(\frac{∂z}{∂y}\right)\frac{∂y}{∂t} + \frac{∂z}{∂x}\frac{∂^2x}{∂t^2} + \frac{∂z}{∂y}\frac{∂^2y}{∂t^2}$
The question at this point is, what is $\frac{∂}{∂t}\frac{∂z}{∂x}$? The point is that (thank you Leibniz notation) it is still true that $\frac{∂}{∂t} = \frac{∂x}{∂t}\frac{∂}{∂x} + \frac{∂y}{∂t}\frac{∂}{∂y}$, so applying this rule, we get:
$\frac{∂^2z}{∂t^2} = \left(\frac{∂x}{∂t}\frac{∂}{∂x} + \frac{∂y}{∂t}\frac{∂}{∂y}\right)\left(\frac{∂z}{∂x}\right)\frac{∂x}{∂t} + \left(\frac{∂x}{∂t}\frac{∂}{∂x} + \frac{∂y}{∂t}\frac{∂}{∂y}\right)\left(\frac{∂z}{∂y}\right)\frac{∂y}{∂t} + \frac{∂z}{∂x}\frac{∂^2x}{∂t^2} + \frac{∂z}{∂y}\frac{∂^2y}{∂t^2}$
$\frac{∂^2z}{∂t^2} = \frac{∂x}{∂t}\frac{∂}{∂x}\left(\frac{∂z}{∂x}\right)\frac{∂x}{∂t} + \frac{∂y}{∂t}\frac{∂}{∂y}\left(\frac{∂z}{∂x}\right)\frac{∂x}{∂t} + \frac{∂x}{∂t}\frac{∂}{∂x}\left(\frac{∂z}{∂y}\right)\frac{∂y}{∂t} + \frac{∂y}{∂t}\frac{∂}{∂y}\left(\frac{∂z}{∂y}\right)\frac{∂y}{∂t} + \frac{∂z}{∂x}\frac{∂^2x}{∂t^2} + \frac{∂z}{∂y}\frac{∂^2y}{∂t^2}$
$\frac{∂^2z}{∂t^2} = \frac{∂^2z}{∂x^2}\left(\frac{∂x}{∂t}\right)^2 + 2\left(\frac{∂^2z}{∂x∂y}\right)\frac{∂x}{∂t}\frac{∂y}{∂t} + \frac{∂^2z}{∂y^2}\left(\frac{∂y}{∂t}\right)^2 + \frac{∂z}{∂x}\frac{∂^2x}{∂t^2} + \frac{∂z}{∂y}\frac{∂^2y}{∂t^2}$
And that's it.