How do you beat penguin?

Use the gadget you get from freeze an get close enough to disable his gun an you got it

You go and rescue Mr. Freeze. When you get to the torture chamber (the middle part of the museum with the frozen water) go to the room on your left side to find Mr. Freeze, you'll have to beat up some henchmen, then free him. He'll tell you to go get a device from his suit and you can use it when you get close enough to Penguin to disable the gun. To find Mr.Freeze's suit you have to go back the way you came, then when you arrive at the torture chamber go straight (When you first entered the room and I told you Mr. Freeze is on the left side, well his suit is on the right side in a room) and go into the room. Beat up the bad guys, and his suit will be in a glass case standing up in the upper left hand corner of the room from where you entered. Walk up to it and you'll take a device from the suit, then go back the way you came and go right, into the Lounge, dodge Penguins shots, and then go through the door way behind him, get close enough to him to activate the jammer, but if you get too close he'll freeze you. After that you punch him off and then you have to fight Solomon Grundy.

Rescue Mister Freeze, and get the Freeze Gun Overdrive. Find the Iceberg Lounge and make your way to Penguin; sliding from "his" gun. Defeat Solomon Grundy by using explosives on the electrical nodes and finish him. Then doge the grenades fired by Penguin to get close and just ATTACK HIM repeatly.