Open-circuit time constant method
You are nearly done.

$$
R_{gd} = \frac{V_xR'_L}{g_mV_{gs}R'_L-I_xR'+V_x}
$$

Use \$V_{gs}=-I_xR'\$:

$$
...=\frac{V_xR'_L}{-g_mR'_LI_xR'-I_xR'+V_x}\cdot\frac{\frac{1}{V_x}}{\frac{1}{V_x}}=...\\
$$
note that those pesky \$\frac{I_x}{V_x}=\frac{1}{R_{gd}}\$. You need to rearrange your equation and get to:

$$
R_{gd}\left(1-\frac{R'}{R_{gd}}-\frac{gmR'_LR'}{R_{gd}}\right)=R'_L
$$

Getting your book result is now straightforward.