Who is the Milfanito by the boss door, and what does she do?

Generally, the Milfanito were created by the "Great Dead One" to comfort the Undead wandering the Shrine of Amana, but they are not the ones who attended the seal that imprisoned the Demon of Song. The few ones left, who once sealed away the Demon, are the hollowed sorceresses that attack you in the Shrine. According to the official guide, they are

Hollowed priestesses formerly responsible for performing the sealing rituals which imprisoned the Demon of Song. Over time the line of shrine maidens gradually died off and, with no successors to pass down the sacred rites, the seal broke and the Demon was freed from its prison. (guide p.253)

The description of the "Demon of Song Soul" reveals, that

When this demon developed a taste for human flesh, it was contained within the Shrine of Amana, but the line of priestesses who looked after the shrine and appeased the creature have died off.

Given that the injured Milfanito is closest to the Demon, it is most likely she has been attacked by the now unleashed Demon of Song. According to the official guide, she is

A scarred and wounded Milfanito found cowering outside the Boss room on the Shrine of Amana, apparently attacked by the Demon of Song. How she survived the encounter is uncertain; she might actually be a ghost. (guide p.416)

Her being a ghost could explain why she vanishes when you try to interact with her. A reason for her being attacked by the Demon of Song could be that she strayed too far into the Demon's territory, without being aware of it. The reason for this is that they

possess only rudimentary awareness of their individual selves; this trait combined with their purity, innocence and general naiveté has resulted in several of their number either wandering into the territory of the Demon of Song or being kidnapped and carried off by outsiders with ill intentions. (guide p.415)

The last part offers an explanation for the imprisoned Milfanito in the Castle's tower, who was kidnapped by the Embedded.

The following deals with who the Milfanito, who the injured one is a part of, are in general. It is possible that they were, along with the Fenito, created by Gravelord Nito, who was the first of the dead.

Both the Milfanito and the Fenito mention that they were created by the Great Dead One. The item description of Nito's "Lord Soul" in Dark Souls 1 reads:

Soul of Gravelord Nito, first of the dead. This Lord Soul was discovered at the dawn of the Age of Fire. Gravelord Nito administers the death of all manner of beings.

The Fenito Agdayne reveals in his dialogue that

I am a Fenito. We weave death, and watch over the dead. This task was granted to me by the one who gave us the first death.

Along with the obvious naming FeNITO and MilfaNITO, this stronly suggests that Nito created both, and they both have the task to look after the dead. The Milfanito in the Shrine of Amana are

songstress[es] given eternal life by the Great Dead One. The Milfanito are an immortal race who sing to comfort those bound by death and Dark, reside in the Shrine of Amana and have never seen the outside world. (guide p.415)

Milfanitos were created to calm the outrageous undead, to stop feelings of rage, lust whatever and turn undead into something like calm angels, like the Demon of Song, but not only..."when we sing the little ones dance" 1) the little ones are the fireflies flying around the creatures on the water in Shrine of Amana, you'll realize some of them jump at you when you open the door and Milfanitos stop singing...cause the fireflies vanished for that instant...you can find this on DS" Wiki "When they sing, the fireflies make the Lizardmen passive, as such they will not attack the player unless they get too close."

Milfanitos were put there by Vendrick to stop Nashandra...has you know (or has you should know) Nashandra wants to reach the Throne of Want and plunge the world into darkness, you can understand her nature by dialogues with other NPC's - she's vile, she gives you tasks to seek the Dragon (for the memories) and the Throne and once you have all necessary items and defeat the Throne Watcher and Defender she comes for you, to get the Throne she desires so much... but too late you're already strong enough to defeat her... the ghostly NPC (Chancellor Wellager) on the stairs of the Throne Room right at the entrance of the Castle says something about a "Queen that came from a faraway land..."

About he injured and caged Milfanitos, chances are they were caught by Nashandra, the injured Milfanito was probably tortured by her, Nashandra can't pass to Shrine of Amanna cause the Milfanitos singing would trap her into a state of relaxation, Nashandra is vile and she can't cross the Shrine...

Don't think Nito has anything to do with Milfanitos...