Your general model for an electric motor could be for example, as follows.
$$J \phi ''(t)=\text{I} \tau -R \phi '(t)$$
with J as your [kg m^2] inertia, I in [A] amps, and R your friction...at the moment, I can't remember the unit is, and $\tau$ your torque constant [Nm/A]
This is a simple linear ODE and is solved also fairly easily.
With initial conditions:
$$\phi (0)=0,\phi '(0)=\phi _0$$
$$\phi (t)\to \frac{J e^{-\frac{R t}{J}} \left(\text{I} \tau -R \phi _0\right)-\text{I} J \tau +\text{I} R t \tau +J R \phi _0}{R^2}$$
This is of course your angle dependant on time, but we want the change, or namely the velocity.
So we take the derivative, $\frac{d\phi}{dt}$
$$\omega(t)=\frac{i R \tau -R \left(\text{i$\tau $}-R \phi _0\right) e^{-\frac{R t}{J}}}{R^2}$$
If we take some a data trace we have from playing the motor at different amperages, we can calculate or fit R and J if they're unknown, like for example:
With the values being {R -> 0.0000186966, J -> 0.000138553} for that particular example.
Or in your case, you can find your torque constant if you know the rest. However, generally you can find that in a motor data sheet...as it appears your don't have one, you can go about it this way. Are you using some kind of software to be able to read such things, matlab, mathematica, or scripted something yourself?
I would highly suggest trying to get a data trace and fit the data to $\omega(t)$ rather than messing around with the maths.
