Chemistry - Can all water soluble ionic compounds conduct electricity?

DavePhD is right! Your material is an organic compound.

In order to be soluble in water, it should have some polar substituents. I'd however exclude carboxylic acids or phenolic $\ce{OH}$ (except maybe in the proximity of a carbonyl group) since these will partly dissociate and yield to minor conductivity.

If the melting point wouldn't be that high, inositol would be a candidate.

The nucleobases thymine and uracil do show melting points in the 300 °C range and are both soluble in water.