Did you try COCOA Butter (Palmer's)?

It says "for Marks, Blemishes and Rough, Dry skin". You can even use it for chapped lips.

COCOA Butter is really cheap, you can find it at any drug or grocery store. Even if it doesn't work, it would not cost you much plus you can use it for other skin conditions. I bought mine for at least 2 years and it still works.

I used COCOA Butter for my heel, which had a big cut. It really worked like charm. I didn't feel any pain the next day. Of course, it may not work for your child. You may want to try it on the hands first before use it around eyes.

Try to avoid using hot water on those areas.

Good luck!