Given: $(5^{\sqrt{2}})^m=25$

Apply the power of a power law to get: $(5^{m\sqrt{2}})=25$

Rewrite $25$ as $5^2$ to get: $(5^{m\sqrt{2}})=5^2$

So, it must be the case that: $m\sqrt{2} = 2$

Divide both sides of the equation by: $\sqrt{2}$ to get: $m = \frac{2}{\sqrt{2}}$

This option does not appear among the answer choices, so we can create an equivalent fraction by multiplying numerator and denominator by $\sqrt{2}$.

When we do so we get: $m = \frac{2\sqrt{2}}{2}$

Simplify to get: $m = \sqrt{2}$

Answer: C