Karate Kid 2 was released on the Amiga (that's the only version I know about I'm afraid)...It was basically a beat-em-up getting progressively harder.
The game tried to follow the plot of the second movie including fights on the bridge, at the house etc...
However, probably the best parts of the game were the bonus levels. These were, as you said, in the form of catch the fly and break the ice. Both included a rather nice digitised picyture...one of Miyagi's head and the other of the scene with the ice blocks in the cafe. Catching the fly was a lot easier with an autofire joystick! As for breaking the ice...an awful lot of joystick waggling was required. The final showdown, as in the movie, was on the centre platform of the old castle. After another fight there was a strange bit where you were supposed to waggle the joystick in time with the drum. I alwys found that if you just waggle away and hit the fire button...you always won!
As well as the figures I was lucky enough to get the dojo aswell. This was really cool. I remember it well. It was mainly red, green and yellow. One big lump of plastic! The figures, as you said, were good. I never had Johnny...just Daniel, Miyagi and a couple of the bad guys from the second film.
Info courtesy of Rich Smith