I looked up the etymology on "kite", and it's quite interesting. Apparently, its original use was to describe a hawk whose call sounds kind of like "kite". Then it was used more generally to describe all birds in that family.

The kite toy was named after the bird, because the bird tended to try to sit on some wind while hunting, so it was basically hovering like a toy kite.

The action of leading an enemy around is named after the toy kite, because a kite in flight is always the length of the string away from the person flying the kite. So, like if you tried to run from the kite, it would always be the same distance away chasing you. You're leading the kite around on a string. And so you would kite an enemy in the same sense.