Answer
See answer below.
Work Step by Step
$\dot{Q}_H=(hA)_H(T_H-T_H^*)$
$\eta = \dot{W}_e/\dot{Q}_H=1-\frac{T_L^*}{T_H^*}$
$\dot{W}_e=\left (1-\frac{T_L^*}{T_H^*}\right)(hA)_HT_H\left (1-\frac{T_H^*}{T_H}\right)$
Defining $x=1-\frac{T_H^*}{T_H},\ y=\frac{T_L^*}{T_H^*}$
$\dot{W}_e=(hA)_H(1-y)x$
$\left (\frac{\dot{Q}_H}{\dot{Q}_L}\right)_{rev}=\dfrac{T_H^*}{T_L^*}$
$\dfrac1y=\dfrac{(hA)_H(T_H-T_H^*)}{(hA)_L(T_L^*-T_L)}=\dfrac{(hA)_H}{(hA)_L}\dfrac{\left (1-\frac{T_H^*}{T_H}\right)}{\left (\frac{T_L^*}{T_H^*}\frac{T_H^*}{T_H}-\frac{T_L}{T_H}\right)}$
Defining $k=\dfrac{(hA)_H}{(hA)_L}$
$\dfrac1y=k\dfrac{x}{y(1-x)-\frac{T_L}{T_H}}$
$x=\dfrac{y-\frac{T_L}{T_H}}{y(k+1)}$
$\dot{W}_e=(hA)_H(1-y)\dfrac{y-\frac{T_L}{T_H}}{y(k+1)}$
$\dfrac{\partial\dot{W}_e}{\partial y}=0$
$y=\frac{T_L^*}{T_H^*}=\left(\frac{T_L}{T_H}\right)^{1/2}$
Substituting back:
$\dot{W}_{max}=\dfrac{(hA)_HT_H}{1+(hA)_H/(hA)_L}\left[1-\frac{T_L}{T_H}\right]^2$
