maybe do some research into pro biotics and get to a health food store and get some probiotic power suitable for kids? Metagenics is the best brand if you can find it.
poor gut function from anti biotics and illness can effect nutrient absorbtion and weight gain/loss.

I'd be doing almond butter, avacado, cod liver oil, salmon, organic butter, egg yolks and coconut cream / coconut oil.

good luck! it can be so stressful having a little one sick and worrying about weight. I have been there with one of mine around that age.