Will distillation do the job of purifying the water?
The old school method of generating ultra-pure water by distillation was to add a small amount of potassium permanganate in a slightly alkaline solution. All the organics were oxidized by the permanganate ion in boiling water. A small of $\ce{MnO2}$ so formed during boiling would take care of the acid(s). This water was good enough for electrochemical experiments. HCl only forms an azeotrope at 20.2% composition and it boils around 110 Celsius.

Today ultrapure water is produced by using quite expensive deionizers with UV lamps. UV destroys all the organics, and then an activated carbon filter absorbs them. Water produced from such deionizer has a specific resistance of $18.2  \rm{M\Omega.cm}$.