The short answer is that, yes, Jesus, the Divine Son, is 100% God (and also 100% man), provided we understand that God does not have “parts” and so could percentages could never be properly attributed to Him.
As the O.P. correctly intuits, the root of the answer lies in the distinction between person and nature.
First of all, it is important to state from the beginning that when we speak about God theologically, we need to be extremely careful not to extrapolate notions that only apply to creatures.
For example, God is utterly simple; He does not have “parts.” (At least, that is the consensus of practically all the Church Fathers, and I think it can be convincingly demonstrated philosophically; such an answer would belong to a different question, however). It is a mistake, therefore, to attribute “percentages” to Him, as if He could be some way be “distributed;” He cannot.
As I mentioned in my answer to the O.P.’s question about the distinction between nature and person, “nature” (or “substance” or “essence”) answers the question, “What is it?” whereas “person” always answers the question “Who?” A person, moreover, is always the subject from which actions originate.
(For example, it is not my arm that pitches the baseball; it is I who pitch the baseball with my arm. It is not my intellect that knows; it is I who know through my intellect. Similarly, the subject of God’s actions are always the Persons.)
Let us analyze the Chalcedonian affirmation “Jesus is fully God and fully man” (in more technical terms: Jesus has the complete Divine Nature and a complete human nature).
In this affirmation, the subject Jesus answers the question Who?—hence it refers to the Person of Jesus. The predicates “God” and “man” answer the question “What?”—in other words, they tell us what are the two natures (or substances, in the universal sense) of Christ.
Now, the Person of Jesus is precisely the Second Person of the Holy Trinity, as was affirmed by the Second Council of Constantinople in 553:
there is but one hypostasis [i.e., person], which is our Lord Jesus Christ, one of the Trinity (Denzinger-Hünnermann, 424).
As I also mentioned in my other answer, each Person of the Holy Trinity is perfectly identical to the Divine Nature (indeed, to the very same Divine Nature, which is undivided and indivisible).
In this sense, it is fairly evident that Jesus (the Divine Son) must be fully God.
So, yes, God the Son is—if you will—100% God. (And so is God the Father, and so is God the Holy Spirit).
This is precisely the meaning of the key affirmation of the Nicene-Constantinopolitan Creed, which affirms that the Son is “consubstantial with the Father” (homoousios tou Patros).*
* The definition mentioned by the O.P. is actually referring to the Incarnation: Jesus, the Son, also assumed a complete human nature at the Incarnation—which makes him fully man. (The union of two natures in a single, undivided Person—also called Hypostasis—we call the Hypostatic Union.)
It should be, however, is that Jesus’ Divine Nature is identical to his human nature. They could not possibly be identical, since the Divine Nature is utterly unique, and cannot be mixed with created natures.
(That is precisely the sense of the famous phrase, quoted by the O.P., that Christ has two natures that are united “inconfusedly, unchangeably, indivisibly, inseparably” in one Person.)