That is amazing! I hope your son keeps feeling better.
On how long you should continue the treatment: I have been taking iodide/iodine for 2 years now, and things are still in progress, still getting better. Think years, not months.
Once things start going wrong in the body, one after another system will get less optimal. At some point things are deteriorated so much that you get major symptoms. But when one finally starts to heal, all those systems will have to be repaired by the body. One after another, in reverse order from how they fell. This takes time.
Medicine works different: most medicine suppress symptoms and because of that, you usually have results pretty quickly. Not that it's doing you any good in the long run, but in the short term, you feel a lot better. Antibiotics is an example of this: it does kill off the wrong bacteria, but it also kills the good. It doesn't differentiate. It is not a solution that repairs what is fundamentally wrong.
Your son will need a lot of nutrients besides Iodine to regain his health. Taking Iodine kick starts the healing process. This healing process will make the body use up his nutrients faster. They have to be replaced. Eat plenty of high nutrient foods. Otherwise the progress he's making now will come to a halt.