Analyzing a MOSFET circuit
You have an equation for \$v_o\$ of the form \$v_o = kv_{gs}\$ but you need \$v_{o}/v_i\$. Use the hint (which is just KVL) to substitute for \$v_{gs}\$: $$v_o = kv_{gs} = k(v_i - v_o)$$ and solve for \$v_o/v_i\$.

For \$R_o\$ you have forgotten the contribution of the dependent current source. To calculate \$R_o\$ set \$v_i = 0\$, apply a test voltage \$v_t\$ across the output, and calculate the test current \$i_t\$ flowing from that test voltage. Then \$R_o = v_t/i_t\$. Since \$v_i = 0\$, \$v_g = 0\$ and therefore \$v_{gs} = -v_s = -v_o = -v_t\$. The dependent current source therefore supplies a current \$-g_m v_t\$. Now use KCL at the output node to calculate \$v_t\$ in terms of \$i_t\$ and re-arrange to find \$R_o = v_t/i_t\$.