I do well taking iodine but each DNA and antibodies are different to the root issues why the disease is present.
If you consider iodine - first a loading test is strongly recommended to see your levels.
Provided you are eating a normal diet you should not need to take extra Iodine.
Whilst a small amount is needed by the thyroid, and this is usually obtained by eating correctly, Iodine in excess can lead to hyperthyroidism.
Iodine is usually not recommended for those of us with Hashi's; however, there are some who take it and do very well. It should not, however, be taken without doctor supervision.
Try asking in the complimentary medicine forum. Someone there might know. I'm not sure, because I don't have Hashimotos. I keep testing "normal", but I've been learning that "normal" doesn't necessarily mean anything for the individual. Someone in complimentary medicine might know. Especially, someone like Paxiled, who has quite a bit of background for where he has his knowledge.