How do I determine the curve of the transmission output on a torque-angular velocity graph for a DC motor?
You can find the answer to the question at Understanding D.C. Motor Characteristics.

What you are missing in this problem is the curve between Torque and Speed that you can find at section 3.1.  

The maximum torque is \$T=8 Ncm\$ (with the gear ratio - \$T_r=160 Ncm\$) and the no load velocity is \$w = 80rad/s\$ (with the gear ratio \$w_r=2rad/s\$). You need to find \$T_f=60 Ncm\$ and the formula is:

$$T_f = T_r - w_f \cdot \frac{T_r}{w_r}$$

where \$w_f\$ is your angular velocity where the torque is \$60Ncm\$