Just before triggering the fight with Madame, Nilin learns that Madame has found a way to turn Remembranes hostile - by way of Remembrane guards attacking her.

Once you find Madame at the main server room, she performs some sort of attack on Nilin that takes her into a Remembrane-like reality. If you have played Batman: Arkham Asylum, think of the Scarecrow sections (only with less platforming and more brawling).

During the fight, Madame will float (often in the distance) in the air, while summoning Remembrane guards to fight you.

Madame has two blocks of health.

Abilities:

Summon Remembrane Guards
Airstrike
Shield
Doppelgangers
Shockwave

Note: Flying Guards/Robots take considerably less damage to kill in this boss fight, so get them out of the way asap!

Phase One:

In the first phase, you must shoot at Madame with your primary fire from The Spammer, while avoiding / taking down Remembrane Guards.

Once you have shot down the shield surrounding Madame, you lose the ability to shoot at her with The Spammer.

To get Madame down to your level, you must use the Sensen DOS attack. This has the added bonus of getting rid of any Remembrane Guards left.

The most efficient way to whittle down her health seems to be to DOS her down, pop Sensen Fury and just go to town.

After you have damaged her enough, she will return to the air, and start bombing the arena in which you fight her. This does not go on for long, but it is a good idea to try to avoid taking too much damage from these attacks. If you get hit, you will get stunned for 1-2 seconds, and lose a full block of health.

Phase One Point Five:

Once you get her first block of health below 50%, she will also start summoning the flying Guards / Robots, which adds another obstacle to getting Madame earthbound.

At this point, Madame will also despawn the enemies, and start bombing the arena again.

Once you remove all of the first block of health, Phase Two will be initiated.

Phase Two:

At the start of Phase Two (with just one block of health left for Madame), she will start out with a new shield which you must shoot away with The Spammer.

After you remove her shield, Madame will use Doppelgangers and create 4-5 copies of herself on the ground.

Credit to [DR/Caribbean]「Shiki」for this contribution to the guide:
During the second phase of the fight, it's easy to identify which is the real Madame. All the images will have their left arms holding the walking sticks; but the clones will all have their right arms on their hips, while the real Madame has her right arm by her side, as shown here: https://guides.gamepressure.com/rememberme/gfx/word/1638540.jpg


NOTE: If you hit 3 Doppelgangers without hitting the real Madame, she will re-spawn the Remembrane Guards, and you will have to shoot her down again.

Once you find Madame, start beating her up. Once she has taken a set amount of damage, she will initiate a Shockwave attack, which you should try to avoid.

Once you have destroyed the Doppelgangers, and Madame has Shockwaved, she will once again summon Remembrane Guards (including the flying guards/robots).

At this point, she will also Shield herself again. However, once you shoot away her shield, she will fall to the ground, despawn Remembrane Guards, and use Doppelgangers again.

Once you get her sufficiently low on health, you will get to the Finisher.

The Finisher:

Once again, a Quick-Time Event (QTE) is upon us. As with other boss fights, if you fail the QTE, you will have to deal damage to the boss in the regular way to get to try the QTE again.

The order I got was:

1. Punch
2. Kick
3. Press E
4. Spam Dodge (Space Bar)

Congratulations!

You escaped from the Remembrane Reality, and have now Overloaded Madame.

You receive 1000 PMP (XP), and the Force Spammer ability, which allows you to move devices "by overloading their control systems and sustaining a constant amount of trash data in their routines".

The Force Spammer ability is used by holding down Shift + Left mouse button, and moving the mouse in the prompted direction.