How can I find the Van der Waals constants (a and b) without any critical values?
I would approach this a little differently.  I would integrate the pressure between $V_D$ and $V_B$, such that the average pressure is 17.76 bars:  $$\frac{\int_{V_D}^{V_B}{PdV}}{(V_B-V_D)}=17.76$$or,
$$17.76(V_B-V_D)=RT\ln{\frac{(V_B-b)}{(V_D-b)}}-a\left[\frac{(V_B-V_D)}{(V_B-b)(V_D-b)}\right]$$
This would provide an equation for expressing a in terms of b.  So the pressure could then be expressed exclusively in terms of b.  Then, to determine the value of the constant b, I would set $P(V_B)$ equal to $P(V_D)$ and solve the resulting equation for b.