jmc

Squaring $x-\frac{1}{x}=i\sqrt{2},$ we get $$x^2-2+\frac{1}{x^2}=-2.$$Hence, $x^2+\frac{1}{x^2}=0,$ so $x^4+1=0,$ or $x^4=-1.$

Then $$x^{2187}=(x^4{)}^{546}\cdot x^3=x^3,$$so $$\begin{array}{rl}{x}^{2187}-\frac{1}{x^{2187}}& =x^3-\frac{1}{x^3}\\ & =\left(x-\frac{1}{x}\right)\left(x^2+1+\frac{1}{x^2}\right)\\ & =\boxed{i\sqrt{2}}.\end{array}$$