Head back to the Zora King, who has been frozen, and go behind him to reach the Zora's Fountain. Go near the place where Jabu Jabu was and jump from one ice block to another to reach the northern side of the map.
Beware: In this place, bottles are really important, even if one bottle may be enough, having two makes things easier.
Go ahead and get rid of the Freezards by swinging your sword several times. Play the Song of Storms in the middle of the room to fill up your hearts and magic, then follow the corridor in front of you. In the ice blades' room, collect all the silver rupees, then get in the next room. There, climb on the ice blocks and go near the blue flame. Take some in your bottles, then use the flame on the chest surrounded by red ice to find the Map. Fill your bottle again, then come back to the previous room. Melt the two ice walls, one on the left and one on the right, then head to the east corridor.
At the end of the corridor, fill up your bottles with another flame, then melt the ice surrounding the chest to get the Compass and also melt the ice to grab the Piece of Heart. Fill up your bottles again, then go back to the ice blades' room and follow the corridor ahead of you. Here, you have to push the square block to collect silver rupees (one rupee is in the hollow right of the entrance). If the block falls into the water, another one will appear. Fill your bottles with the flame, then to reach the upper corridor, make a new block appear. Push it left (from the entrance's point of view), then towards the flame, right, towards the entrance and right (towards the corridor). Climb on the block and go on. Melt the ice on your way, then open the door. Fight off the White Wolfos then pick up the Iron Boots in the chest. Then, Sheik will teach you the Serenade of Water. Get out of the Ice Cavern with at least one bottle of blue fire. (You can use the hole filled with water in this room as a shortcut, using the Iron Boots). Go back to the Zora King and melt the ice surrounding him. Talk to him and he will give you the Zora Tunic. Warp to Lake Hylia using the Serenade of Water. Before this, you can also melt the ice blocking the Zora's shop.
Wear your Zora Tunic, then get in the water and put on your Iron Boots to stay at the bottom of the water. Fire your Hookshot at the blue diamond, get in, go straight ahead and take off your Iron Boots.