For a string fixed at both ends, the amplitude of a standing wave at a distance $x$ from a fixed end is given by $A(x) = A_0 \sin(kx)$, where $A_0$ is the amplitude at the antinode. In the second harmonic ($n=2$), the length of the string $l = \lambda$ (since $l = n\frac{\lambda}{2}$). Thus, the wave number $k = \frac{2\pi}{\lambda} = \frac{2\pi}{l}$. Given the amplitude at the antinode $A_0 = 2\text{ mm}$. The amplitude at a distance $x = l/8$ is: $A(l/8) = 2 \sin\left(\frac{2\pi}{l} \times \frac{l}{8}\right) = 2 \sin\left(\frac{\pi}{4}\right) = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}\text{ mm}$.