Find utility function given indifference curve?
VCG's comment about isolating $k$ is the correct approach.

Given $U(\cdot) = k$ and
$$z = \frac{k^\frac{1}{\delta}}{x^\frac{\alpha}{\delta}y^\frac{\beta}{\delta}}$$
raise each side to the $\delta$ power:
$$z^\delta = \frac{k}{x^\alpha y^\beta}$$
and isolate $k$
$$k = x^\alpha y^\beta z^\delta$$

Since $k$ is now a function of each of the goods in your equation, it makes sense as a function form for utility.

$$U(x, y, z) = x^\alpha y^\beta z^\delta$$
So we arrive at the Cobb-Douglass utility form.