The problem with Antibiotics is that it doesnt work long term; what I see is that the infection tend to come back, time after time, one Antibiotics lead to the next etc.
What you could try was to double the iodine, or even triple it. Then make a solution / eye drops of borax, destilled water and a drop of iodine, then two as suggested by HighOnWater.
Borax kill staph, is non stingy for the eye, sodium (salt)based. Together with Iodine I cant see this last for long.
I also suggest loads of vit C with any infection.
I have used borax-water alone for kids eye infections, or when irritated eyes (red,burning)-it works great.