Hi - at that age I wouldn't use too much water or shampoo. I wet a washcloth and wring it out. Add a small dab of shampoo and just rub it around the hair. Even with minimal water/shampoo, there is enough lather to clean hair. I then rinse the cloth with water and wring it again, then wipe off the shampoo front to back. With this minimal water/shampoo nothing should be running into eyes or elsewhere at all. Then use a clean washrag wet with clean water to do a final wipe on hair and wash off any remaining shampoo.
Since you are using the sink, does that mean you are using the sprayer? If so I would avoid that and use method above. I think it is too much water for this age. Good luck!