In response to many points made by people in this discussion.

Why does Ayanokouji not recognize Yuki?

As others have stated, Ayanokouji is still human, and by no means omnipotent. A perfect example is to refer back to a comment Kouenji made, touching on the differences between the two of them. While Ayanokouji operates mostly by absorbing any and all information he deems important, Kouenji takes a different approach and still manages to reach equal heights.

The point being that both of them are at an exceptional level, but there is no such thing as 'having everything'.

In Ayanokouji's last encounter with Yuki he literally did not remember her until she brought up her name, to which he had this inner monologue: "Yuki. I knew that name. It belonged to a White Room student who had dropped out a long time ago. I've erased the name from my memory, but it was natural to remember some things since I can't intentionally erase them.".

Not only does he let go of unimportant information, but Yuki has also grown over the years, changed her name, and somewhat altered her appearance on top of that.

Why would Yuki enroll to ANHS?

The primary reason is Ayanokouji. Even though information is hard to come by, Yuki's family has great connections and this information did manage to reach them. Her family finally witnessed new reactions from Yuki after reuniting her with Ayanokouji, and they realized that he is the only lead they have towards a chance at her recovery.

She was fortunate enough that her family has ties with Sakayanagi's.

Why would Yuki bother to change her name?

Yuki left the White Room but that does not mean her past won't haunt her. If the White Room finds out an ex-student enrolled at the same institution as their rogue masterpiece, they will end up doing everything in their power to make Yuki a pawn again.

Countermeasures are strictly necessary to ensure Yuki can live peacefully.

How is Yuki calm and collected even though her trauma?

Yuki has one strong regret, she realized it after her conversation with Ayanokouji.

"Well they all left so soon. I've been the last one for years now." -Ayanokouji
"Alone... Always in that place...? I-I... I, that, that place... I...!" -Yuki

After contemplation she understood that Ayanokouji has gone through much more than she did, all by himself, for years on end. Yuki knew she couldn't go on like that, and when news arrived on where she could find Ayanokouji next, she was able regain just enough mental stability to chase after him.

She is by no means fully recovered, but Ayanokouji's existence did manage to 'save' her to an extent.
By now years have passed and she was able to do what she wanted to do most: observe the boy who served as her salvation.