Replacing Electrical System with a Gain?
The electrical system can be described by \$V=iR+Li+αẏ\$, where α is the
  relationship between the magnetic field induced voltage and the
  velocity.


That may be true but, the TF of the electrical system within the dotted line can also be described as: -

$$\dfrac{i}{ij +s}$$

Where i and j are the gains of the two blocks used and shouldn't be confused with the complex operator i or j.

In other words this is a single order low pass filter with gain. Can you take it from here?