I find that songs sung by Harry Champion often contain a lot of slang expressions that are no longer current. Some, but not all of it, is Cockney rhyming slang. That's one of the things that make his recordings so hard to understand today, especially for an American. (That, plus the quality of the recordings and the fact that he sang so fast.)

Maybe "kite" meant something different to him than it does now.

Changing old songs so that they make more sense to today's audiences is one of the common ways songs evolve during the "folk process."

Personally, I'm more interested in documenting what Champion actually sang, than in deciding what a singer ought to sing today. In this case, I'm pretty sure he sang "just like a kite."