Can $g_m$ be solved for? Or do we need to know $i_d/v_{gs}$ in the saturation regime?

Well, $g_m$ is a property of the FET- it will be specified by the FET manufacturer. So if you have to design a circuit for a particular FET, this will be a fixed parameter.

Now if you have flexibility in designing a circuit, you might want to choose a FET with a particular $g_m$ (although note that it can vary from FET to FET of a particular type) to make an amplifier with particular properties.