Factor the numerator of the first fraction and the denominator of the second fraction.

The numerator $2m^2 - 2m$ can be factored by taking out the common term $2m$.
The denominator $m^2 - 2m + 1$ is a perfect square trinomial, which factors as $(m-1)^2$.

$$\frac{2m^2 - 2m}{m} \times \frac{1}{m^2 - 2m + 1} = \frac{2m(m-1)}{m} \times \frac{1}{(m-1)^2}$$

The $m$ in the numerator and denominator of the first fraction cancel out.

$$= 2(m-1) \times \frac{1}{(m-1)^2}$$

$$= \frac{2}{m-1}$$

The simplified expression is:

$$\frac{2}{m-1}$$