Hi- I have been in the furniture business for about 33 years. First I would contact the store where you purchased the chair and tell them that the recomended cleaning method of distilled water is not working. They may have some suggestions for you, or a cleaner from the manufacturer. Most of the time leather can be cleaned with a moist towel, if that does not work most leather companies suggest a mild soap like ivory. Mix it up with water-like you are going to wash dishes-put your soft cleaning cloth in the soapy water. Wring it out until it is just moist (you don't want to make a sloppy mess!)and wipe the chair. Follow it with a damp with just water soft cloth, and then wipe it with a dry cloth. Most leather has a finish that will take this kind of cleaning. Some leather is oil impregnated and you can not wash it in this manner. Test what you are going to do in a hidden spot and wait a few days to make sure the leather is not having any reaction--rare, but sometimes it does happen. Hope this helps!
Pam