Have you heard of the iodine patch test? You get iodine liquid in a drug store and paint on a 2 inch square patch on your skin someplace where it won't be washed off within 24 hours. If that square fades completely within 24 hours than your body is low in iodine. If it is not low in iodine then your body will not absorb it completely in 24 hourse.
Some have said that this test may not be accurate, but I don't remember any explanation. I think it might be useful instead of taking supplements and not really knowing if you need them.
This site says that kelp will aggravate Hashimoto's (in the first question).
http://www.thyroid.ca/Guides/HG13.html#1