Why is distilled water such a poor conductor of electricity?
You are mistaken.

Water is not composed of these ions. There is ongoing autodissociation equilibrium reaction$$\ce{2 H2O &lt;&lt;=&gt; H3O+ + OH- }$$ shifted strongly toward left, with concentration of ions  $\pu{1e-7 mol/L}$ at $\pu{25 ^{\circ}C}$ . 

Therefore conductivity of water itself is very low.