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

$=\frac{1+i^{2}+2 i-1-i^{2}+2 i}{1^{2}+1^{2}}$

$=\frac{4 i}{2}=2 i$

$\therefore\left|\frac{1+i}{1-i}-\frac{1-i}{1+i}\right|=|2 i|=\sqrt{2^{2}}=2$