The Iodine used for painting is Lugols 5 % It dries quickly and stick to the skin for some hours.
How much; just a drop will do it; if the child can hold the jaw open for a few seconds after painting it should be ok.
The problem is the young age, not easy to make a two year old cooperate this way.
Alternatively put him on higher iodine, like 10 drops; and high vit C, selenium etc.
Also painting on the outside skin, under the jaw, may help.