I have an older nutrition book, from the 1950s & 60s, when Iodine was still remembered as a good all-purpose medicine. Anyway, the author states:
"If Vitamin E is deficient, the absorption of Iodine by the thyroid decreases to 5 per cent of normal...Giving adults 500 milligrams of vitamin E daily has caused their thyroid glands to absorb twice as much Iodine and the protein-bound iodine in their blood has increased to normal. Individuals with toxic thyroids have also been helped by taking 4 - 6 mg of iodine daily."
However, I do not know if they are talking about Iodine or Iodide. Or both.
Also mentioned is the necessity of consuming either a high protein diet to supply the amnino acid TYROSINE, from which thyroid hormone is made. Tyrosine, however, can't be used without vitamin B6 and C or cholin. Thyroxin itself is inactivated by oxygen if vitamin C is undersupplied or destroyed.
Nowadays, however, you can buy Tyrosine in capsule form at health food stores.
So, there you go. Maybe there is something useful for you here. It is worth a try!