Unipolar transistor's transconductance/admittance
The relation between transconductance (\$g_m\$) and the conduction parameter (\$K_n\$) may be found from the definition of \$g_m\$:

$$
g_m = \frac{\partial i_D}{\partial v_{GS}} = 2K_n\left(v_{GS}-V_{TN}\right) \\
\implies K_n=g_m/\left(2\left(v_{GS}-V_{TN}\right)\right)
$$

In other words, the conduction parameter may be found from the transconductance when given the overdrive voltage at which \$g_m\$ was measured.

It is also possible to find \$K_n\$ from \$g_m\$ when given the drain current at which \$g_m\$ was measured:

$$
g_m = 2K_n\left(v_{GS}-V_{TN}\right) = 2\sqrt{K_n}\sqrt{K_n\left(v_{GS}-V_{TN}\right)^2} = 2\sqrt{K_ni_D} \\
\implies K_n=g_m^2/\left(4i_D\right)
$$

In practice, the conduction parameter may be useful for back-of-the-envelope, or approximate, calculations.

Reference: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-012-microelectronic-devices-and-circuits-fall-2005/lecture-notes/lec11.pdf.  See slide 11-6 and apply definition \$K_n = W\mu_nC_{ox}/\left(2L\right)\$.