Only fallen human nature... and even then, it is not a matter of defeating human nature (how would Christ instruct us to defeat something that he assumed in its fullness?), but rather working to make complete its transformation -- a transformation that begins with our baptism when we undergo the three-fold immersion in the saving waters.
When we undergo the spiritual labors that the Orthodox Church points us towards, we are not escaping or defeating human nature. We are rather working to become truly and fully human. This is why when we pray certain Orthodox private prayers, we ask God to deliver us from "animality." Ultimately, the choice is between becoming transformed into the glorified human nature (which participates directly in the Divine through the energies of God) or becoming like the animals. Or, as another prayer puts it "worse than a beast."
There is a very real sense in which, in its practical theology, the Orthodox Church does indeed understand and take into account bhfallen human nature. The canons, the commandments of the Gospel, the prescribed spiritual and physical labors, and the prayers of the Church are necessary because we, in our fallen human nature need them. It is because of our fallenness that we have, for instance, the fasting guidelines of the Church. If we were able to be temperate and continent, we wouldn't need such guidelines. And it is only within this context that Orthodox guidelines on relations within the marital bond are to be understood.
I'm surprised that the discussion of artificial contraception has gone on this long on this thread. Only an Orthodox priest who is well-versed in the spiritual life has the grace to deal with this subject, and even those usually won't speak on it in public, because each individual Orthodox Christian needs to be met where he is in the spiritual life. It is (like our teachings on the Theotokos/Panagia and on the inner spiritual life) not a part of the public witness of the Church. That witness is, and should be, largely restricted to the teachings of the Gospel.
Is there any other kind in us?
but rather working to make complete its transformation
From what to what? Animal to human or human to god-like? If we go from animal to human, we go from innocence to corrupt; if we go from human to god-like it is not becoming more human, but more unlike what we are. We are not animals, not because we are better then they are, but because the animals are not fallen. We are insulting God's creatures when we say "animal like."
You say that God took our human nature -- yes He did, so he could die for us. Even God, Who became Flesh without sin, and lived a life without sin, even He was mortal in His human form (although our teaching says: that which is without sin is immortal). The Son of Man suffered and died on the cross; the Son of God neither suffered nor died.
Te whole aim of Orthodoxy is to help us humans become something better by the process called theosis. The Orthodox standard, as you say of "a pious ascetic monastic saint, living 'the angelic life'" is anything but what our human nature tells us to do. It's a struggle, not some natural process; it's a transformation -- metamorphosis, a new form of life; it's an act of conscious self-denial, not something that comes "naturally."
There is nothing, nothing in the teachings of Jesus Christ that even resembles anything the Church made later on with respect to humanity, especially human sexuality, and veiled it in some kind of psuedo-theological exercise of personal preferences.
When we read that someone in medieval Russia took the time to count the number of times a woman came to church in a month, then accused her of being "unclean," and the church condoned that by punishing her by denying her the Eucharist for up to three years (!), that is not knowing human nature but sheer ignorance, prejudice, and superstition. It is not the business of the Church to tell you how to live -- but to teach you how you should live. After all, even God let's us make a choice whether we turn to Him or whether we face the darkness; bat ism in and of itself does not change your nature; you are still prone to sin and, left in the dark, will choose that which your nature compliers you to do -- and it will never be theosis.
Either we do it out of love or we fake it. I don't fast because the Church says I should. The Church has changed its mind on fasting a number of times (yes, the primitive and the Orthodox Church; never mind the Catholic!). I fast because I want to, not because I have to. It gives me pleasure to fast. Yet my nature tempts me to do otherwise.
So, do we still count the number of times women show up in church or did women somehow through the ages become "cleaner" when they are on their period??? Could it be that the Church was (a) ignorant or (b) wrong or (c) both -- to do so? Or do we simply choose to sweep the whole embarrassing episode under the rug and hope that no one will ever discover it? Oh, let me guess -- we choose not to discuss it in public, as you so clearly point out.