Surely the most shocking feature of this narrative is that Nebuchadnezzar did not call in Daniel to interpret his dream. In verse 6 of chapter 4 we find him calling for the wise men of Babylon and getting a very similar grouping to the one he received in Chapter 2. I think the key to seeing why this is actually comes from an often-overlooked feature of chapter 3. Whilst Nebuchadnezzar had a growing respect for the Hebrew God he didn't necessarily like him and possibly even thought he could avoid him.
When Daniel interpreted the dream of chapter 2 for Nebuchadnezzar, Nebuchadnezzar described God as 'God of gods' and 'Lord of kings' but it is probable that that expression was either hyperbole or a short-term emotional reaction. The expression he also used was a revealer of secrets. I think it quite likely that this expression was quite a genuine one. Babylon was polytheistic; they were quite able to accept the concept that different gods had strengths in different areas. So in Nebuchadnezzar's mind he had pigeon-holed the Hebrew God and could now carry on with business as usual.
Nebuchadnezzar's real heart based response to the interpretation is given in Chapter 3. In chapter 2 Nebuchadnezzar was told of an image of four metals, the head of gold was Babylon and the chest of silver was the kingdom to replace the Babylonian one. So he built a huge statue, all of gold. He agreed with the idea of him being the head of gold, he didn't agree with the idea of his kingdom being replaced.
In chapter 3 Nebuchadnezzar's understanding of God was broadened further, He could preserve his own. Again though the precise text is telling. In verse 28 he refers to "the God of Shadrach, Meshach and Abednego". Nebuchadnezzar had accepted that God could reveal secrets, that he would protect his servants and even that Nebuchadnezzar and the rest of Babylon should not attempt to fight against God or his servants.
However, when it came to an issue that troubled the king personally he fell back upon the wise men of Babylon whom he expected to give him the interpretation he wanted. Verse 7 shows that exhibiting the same behavior produced the same result, the wise men were unable to produce the interpretation. It is interesting that this time he revealed the dream to make it easier for them and they were still unable to produce a result.
As one of the wise; men Daniel eventually showed up. Why the other wise men had to exhausted themselves before he turned up we don't know. Possibly it was God's doing, he wanted Nebuchadnezzar to know this was the only interpretation. Possibly it was Nebuchadnezzar that wanted any interpretation he could get other than Daniel's.
Whatever the driving force behind it, it is clear that once Daniel had shown up Nebuchadnezzar was respectful to him and ready to hear the interpretation that Daniel gave. Verse 19 is also interesting in that it suggests that Daniel also had some form of respect for Nebuchadnezzar. We also see that Nebuchadnezzar was sufficiently keen to hear the interpretation of the dream that he was will to brace himself for an unfavorable one.
There is undoubtedly a lesson in here for us. How often do we repeat secular practices that we know don't work when simple dependence upon the Lord would solve the problem? If the answer you give is 'very' then you next have to ask 'why?' Is it simply that we forget, or are we, like Nebuchadnezzar trying to get a result that we think we may like rather more than the one that God will give us?