The follicle is about 1/4" inch deep. I would say you should see regrowth in 3 weeks if it's going to work. I would look into Youth-if-I (or similar) hair vitamins and Nigella sativa (aka black seed) for its autoimmune help. You want to give the hair follicles all you can to support them....
Hair dye can be dangerous and cause permanent damage to the hair roots causing permanent hair loss when it is not used properly. There are warnings labeled on all such products