The Bronze Mirror is an essential item which allows you to use Backside Doors once the Guardian of a field has been defeated. Though it is located in the Temple of the Sun, obtaining it requires solving puzzles in both the Spring in the Sky and Chamber of Extinction. To those who played the original, the Bronze Mirror is no longer obtained by defeating Ellmac and can actually be obtained beforehand.
Begin by following the steps for Flooding the Temple of the Sun. When you're finished, you should be looking at the following:
Jump down into the opening to enter the Temple of the Sun (F-3). Be mindful of the Sphinx and proceed to (G-4), which should now be flooded. A jet of water will rise out of the vent in the floor, allowing you to reach the Origin Seal. Breaking it will open the path to the right, allowing you to enter the Chamber of Extinction. This area will be dark, so follow the image below to find your way. If you have the Flare Gun, you can fire a flare at the torch just above the tablet to temporarily light the room. Attack the circled block from the left until the Shell Horn plays, then make your way back out to Temple of the Sun (G-4). Note that if you made your way through the top path, you cannot attack this block from the right side.
There should now be an opening (1). Jump into the small area and break the pots inside. Breaking the upper left pot will reveal a dais (2). Place a weight here to unlock the chest (3).