According to Islam, Prophet jesus (peace be upon him) does not have divine qualities. Prophet Jesus was a man and a messenger of God. In being called to be one of God's prophets, he was clearly elevated to a very high status, but he was, in the end, quite simply a man. This is very clearly the Muslim teaching about Prophet Jesus.
To be fair to you, though, we can elaborate more in giving a fuller answer. Perhaps the best way is to state very clearly what Islam teaches about the Bible, since you refer to the Bible, then to state very clearly what Allah Almighty says about Jesus in the Quran.
Muslims believe that the only certain record we have of Allah's words to mankind is to be found in the Quran. The Quran is His final message to humankind, intending for all people, everywhere and for all time.
You see, the previous messages He sent through other prophets such as Abraham, Moses, David, and Jesus (peace be upon them all), were intended for a particular people at a particular time.
[Indeed We sent down our messengers with the clear signs, and We sent down with them the book and the balance, so that people may uphold justice.] (Al-Hadid 57:25)
And again we read:
[Indeed! We sent a Messenger to every nation.] (An-Nahl 16:36)
For the very reason, though, that they were not meant to be for all people and for all time, Allah allowed these previous books, such as the Torah and the Injeel, to lose their precision over time.
In other words, through repeated translation into different languages, or by deliberate alteration, these books can no longer be believed as the exact word of God (Allah). For example, in the Quran we read:
[Some of the Jews pervert words from their meanings.] (An-Nisaa' 4:46)
And also:
[So woe to those who wrote the Book with their own hands, and then say: "This is from Allah," that they may sell it for a little price. So woe to them for what their hands have written, and woe to them for their earnings.] (Al-Baqarah 2:79)
In other words, then, we cannot be sure about the truth of what is in the Bible. If it agrees with what is in the Quran, we can accept it as true. If it disagrees with the Quran, we can accept that it is wrong. Where the Bible says something which is not in the Quran, we do not know if it is true or not.
For people to say, then, "it says in the Bible…", Muslims have to ask if it says the same thing in the Quran. If not, it is not a part of our faith. The only place we have any mention of Jesus having Divine qualities is in the four Gospels chosen out of many by the early Christian Church.
Written after the event, and heavily influenced by the writings of Paul, these Gospels were written for believing Christians. They were not written as proofs of anything, but as affirmations of what these early Christians believed.
Using the Gospels as proof that Jesus had Divine qualities does not work for Muslims, since the Quran says very clearly that he was not:
[Christ the son of Mary was no more than a messenger; many were the messengers who passed away before him.] (Al-Ma'idah 5:75)
And also:
[And they take as their Lord, Christ the son of Mary; yet they were commanded to worship but one God: there is no God but He. Praise and glory to Him. (Far is He) from having the partners they associate (with Him)]. (At-Tawbah 9:31)
So what else do Muslims believe about Prophet Jesus? Well, they believe that he was born of a virgin mother and spoke from the cradle (Aal `Imran 3:42-47); and that he performed miracles (Aal `Imran 3:49). He was related to God only in the same way that Adam, the first man was:
[Truly, the likeness of Jesus with God is as the likeness of Adam. He created him of dust, and then said to him, 'Be!' and he was.] (Aal `Imran 3:59)
A very important distinction between Muslim and Christian belief concerns the nature of Jesus' death. As you know, it is central to Christian belief that Jesus died on the cross and after three days was raised from the dead.
This is important to Christian teaching because Jesus is seen as savior of the world. His death on the cross, as God's Son, is seen as the way in which God forgives the sins of the human race.
Islam is totally different. First of all, we do not believe that human beings are born in a state of original sin, nor do they need a savior. In fact, as Muslims we believe that people come into the world in a naturally pure state. So what would a savior save them from?
As important as this, is what the Quran says about this alleged death on the cross. Talking about Jesus the Quran says:
[But they killed him not, nor crucified him. For of a certainty they crucified him not.] (An-Nisaa' 4:157)
In other words, Jesus did not die on the Cross. This is quite a major difference between Islam and Christianity.
So, in answering your question about separating the Divine and human qualities of Jesus, as Muslims we are perfectly clear. Jesus did not have Divine qualities.
He was a man, as all men are, but he was called as a man to be a messenger of Allah Almighty.
Allah revealed a message to him, but through time the original words of this message have become lost, and what remains is a distorted version which elevates Prophet Jesus to the rank of Deity.
It is an absolute of Muslim belief that Allah is One, without partner or equal. Jesus was called, like all prophets before him, to proclaim that message to all mankind.
[He said; "I am indeed a servant of Allah: He has given me revelation and made me a prophet: and He has made me blessed whosesoever I be, and has enjoined on me prayer and zakah as long as I live."](Maryam 19:30-31)