Did Nina die?
Yes and No - it doesn't say whether she died which is what makes it perfect.
It simply says - Realize you desires in life much much before you lay on your death bed. In other words, convert to a black swan.
From one perspective, Nina didn't die but converted from White Swan to Black Swan (fading white light). Further explanation given below in layman terms.
In my opinion, the movie is full of personified characters:
Teacher: Teacher of life or mind. It may appear that he's portrayed as lustfully aggressive, but in the context of what the writer wants to project, he is a positive character simply because the lust (in the given context) symbolizes protagonist's desires in life which is what the teacher is trying to uncover and help her get to her (positive) dark side. When he says "Go and touch yourself, that's your mini home work for the day", he means "Introspect yourself"
Black swan: unfulfilled desires many realize towards the end of their lives. In the movie it is shown as lust. Lust, here, is symbolized as positive desires which is what the teacher tries to encourage his students about. The teacher says "open your mouth", "lose yourself" etc. Teacher (of life/mind) suggests her to open up and let herself free and enjoy life rather than staying in her own shell/cell of restrictions/social attention/anxiety/fear etc.
At the end the protagonist does open her mouth willfully - indicates she has opened up in life. In the middle of the story, the teacher gives her a home work and she does it with passion, which shows she does internally have the dark side but is unwilling to open up.
White swan: represents closed minded futile innocence in us. Many a times, people live just for society portraying themselves as innocent albeit they have an inner darker side. The only time they realize this is on their death bed (recollect the last seen). It takes blood/mirror (introspection/self-realization) out of you to visualize what you actually need/needed in life, but phew! there's only one life. Reality gives you only one life and you can't switch to another unless you realize before your death and become a black swan. The white swan (unnecessary innocence, hesitancy, fear of society) has to die for completing your life (black swan) and making it perfect!
Death/fading white light: Death of innocence and emergence of her dark side. To be perfect in life, we need to lose ourselves and let loose our dark side without fear of society
Last scene: In life, we fluctuate between innocence and our inner dark side. As soon as the inner mirror (shown as glass mirror embedded in her blood) comes out (stands for introspection), she starts enjoying life and performs her best. The white swan i.e. unnecessary innocense (indicated by her white dress) dies and black swan emerges. This is indicated by the Teacher acknowledging her with a smile when she says - "I've felt it, I am perfect"
The movie also says - Do not focus into others or your competitions or politics (stands for challenges in life) but just focus on yourself and you win. The only thing that can stop you is yourself!