Explain why ionic compounds such as potassium sulfide have high boiling points and conduct electricity when dissolved in water?

Such compounds have high boiling points because bonds between the atoms are strong. The boiling point largely depends on electronegativity difference between the atoms - as this difference is greater, so is the force that keeps them together. They conduct electricity when molten because ions are free to move in liquid state, due to weakened intra- and intermolecular forces, i.e. ions begin to separate as they gain more energy, be it through electric potential or heat.