Once you managed to get in the first room of the Eagle Labyrinth, go in the room to your right.  Take out the stalfos in that room to obtain a key.  Then go to the room to the left of the first room to take out the keeses for the other key.  With that settled, return to the first room in the labyrinth.

Unlock the north door.  Prepare to take on Stalfos.  If you got bombs, bomb the north wall for a secret door.  Before going through the newly made door, go east and get the compass.  Then go left past the room where the secret door had been created to take the map.

When that is settled, return to the room where the secret door had been bombed out; else go north from the room with the key.  On the other side of the secret door is some red Goryias for the map.  Continue north and eliminate more Goryias.  Grab the key and go left.  The room has traps -- set the traps off and get out of the way before making your move.  (The easiest way to accomplish this is to stay out of the way until both traps have actually met and started to retreat back to their original locations.)

Make it to the other side of the room and push the farthest left block to open the secret passageway.  Go down the passageway and grab the Bow.  Once the Bow has been grabbed, return to the room where you found the map.

From that room, go right and take out the Goryias for the Boomerang.  The next room, if you got at least one key, avoid the Wallmasters and go north.  Else, wipe out the Wallmasters for the key and then go north.

There, you will battle Aquamentus.  If you got bombs to spare, use them.  Else, attack him with the sword.  If you wipe him out, the Heart Container appears.  Grab it and go to the right room for the Triforce fragment.  When you grab the triforce fragment, your health is restored and you are warped out of the Eagle Labyrinth.