Climb up the stairs where you came in and go through the locked door.

Kill the eye laser thing with a bombchu, then go through the next door and kill that eye laser thing with a bombchu (it will go down the wall, across the floor, and up the other wall to get it).  Kill the weird floating guys with fire arrows or try luring them into the fire that appears when you step on the switch.  Go through the door that opens.  Ignore the statues and go stand next to the closed door to your right.  Equip normal arrows and fire one at the statue furthest from you.  As it moves toward you it will step on the blue switch which opens the door, so quickly go through it.  Go up the stairs (there's invisible chests in here with hearts in them) and through the door.  You'll have to fight another Iron Knuckle.  This time is a little different since you're now older Link.  If you use the Big Goron's Sword you can stab by pushing forward+A and then quickly retreat to get in hits before he hits you.  Using this sword will kill him in a lot less hits then it took young Link to kill him with.  If you don't have the Big Goron's Sword wait until after he's done swinging his axe and then hit him.  Once he's dead go through the door to end up outside, and this time the chest holds the Mirror Shield.

Go back inside and down the stairs to the room with the four statues.  Stand in the sunlight and use your Mirror Shield to reflect the light on to the sun symbol on the wall.  Go through the door that opened and get the key from the chest.  Go back out of this room and then take the door to your right back to the room where those weird floating guys were.  Go through the locked door here. Kill the two eye laser things with Z-targeting and a couple of Bombchu's (BTW your new Mirror Shield can now block their lasers!).  Shoot down the skulltula halfway up the wall.  Now aim your hookshot at the climbable section of wall at the very top and fire it.  You'll be pulled up to the top climbable section rather than having to navigate this moving wall.  At the top you have two more eye laser things to kill then go through the door.  Play Zelda's Lullaby on the Triforce symbol to open the door in front of you, then go through it.  All the doors in this room are fake.  Blow up the one just to the left of the chest to reveal a gold eye.  Shoot it and another block of ice will appear above you.  There's a hookshot target on the ceiling above it to use that to get up onto the new ice block, then jump over and step on the switch.  Drop down and get the Boss Key, then exit the room.

Turn right and go through the door.  Watch out for flying pots, and hit the switch behind the grating with a spin slash.  Go through the door that opens and kill all the baddies in this room.  You can use the Mirror Shield to reflect the light onto a sun symbol above the archway between the two sections of this room to get some bombs.  Blow open the wall to the statue's right then push it around so the light shines down the new passageway.  Go down there and push the second statue so the light shines back into the previous room through the grating.  The face will crumble off and a grate will be behind it.