If the Bible contains all of the word of God and there can be no more, then why do we need the Nicene Creed?

Something that the answers above did not cover is the very strong assumption made in the title regarding the scope of the word of God, which is important for the question itself. Established Chalcedonian Churches would not make any claim of the form "all of the word of God is contained in..." or "there can be no more". Indeed, the two most prominent Churches, Catholic and Orthodox explicitly reject that view. For a very recent and clear explication of the Catholic view on the matter, see the Verbum Domini here: http://www.vatican.va/holy_father/benedict_xvi/apost_exhortations/documents/hf_ben-xvi_exh_20100930_verbum-domini_en.html

Word of God, strictly (and capitalised) refers to Christ, an eternal an inseparable part of the Trinity, as established in John. Naturally, there is no sense in which the Bible can "fully contain" Christ, and a fundamental precept of the Christian fate is that the Word is eternal, not only can "there be more", but there is. The Christian tradition, explicated in human words can be referred to as "word of God", and frequently is referred to in such a way, but the reference, for orthodox traditions, is one by analogy and to signify that particular human words announce, contain or explain the Divine truth. It's worth noting that "theology" is, literally, "word of God", and that was the only meaning of theology in the Christian world up until the time of the scholastics, when it started acquiring the more modern meaning of a rational study of religious concepts — and scholastics certainly would have found the statement "there can be no more theology" very strange! Generally, by orthodox doctrine, the words which explicate the Divine truth are not limited to Scripture, the exhortation linked above is clear on the topic: "the word of God precedes and exceeds sacred Scripture". This, of course, is not to detract from the singular status that Scripture has within the orthodox traditions.

This should help you contextualize the use of the Nicene Creed, which, as correctly answered above, is not a supplement to the Scripture, but a concise and clear re-statement of the truths already contained within the Scripture. Why is there a space for such re-statements? Exactly because the word of God is a living tradition, a tradition which ensures our continuing comprehension of the Holy message expressed in the Scripture. And here 'expressed' is a far better word than 'contained' — orthodox traditions do not claim divine authorship for the Gospels, they are human works comprising human words, for all words are human — it is the divine inspiration behind the words that gives Scripture its singular status.

There is no contradiction between saying the Nicene creed seems to be a reasonable summary of many of the things the Bible teaches, and accepting nothing just because it is in the Nicene creed, but justifying every belief based on Bible texts.

There is a danger that a creed will be treated as if it were scripture. It is very important that Christians test every doctrine against the Bible. Doctrines that have been discussed and no evidence found to overturn them after many centuries, and by Christian groups with widely differing beliefs in some areas, cannot lightly be changed. Nevertheless, no creed is scriptural, and no Christian should be accused of doctrinal error if they differ from it, provided they can show Biblical evidence for their belief or at least that the Bible does not conflict with their doctrine.

Some branches of Christianity do have fixed doctrines that they seem to be unwilling to put to the Biblical test. Perhaps in some cases they are treating their creeds as if they were scripture.

I define Scripture as those books of the Bible upon which the greatest part of the whole company of Christian believers agree to be divinely inspired word of God.

The Nicene Creed is not an "extra-biblical doctrine", but rather a concise statement of a summary of the essential parts of the Christian faith. The most fundamental reason for the Nicene Creed (and the same applies to the Apostle's Creed, though that's not part of the original question) is that it is impracticable to recite the entire corpus of Scripture as a statement of faith and is a parallel to Jesus response to the Pharisees asking about the greatest law in Matthew 22:37b-40:

Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. This is the first and great commandment. And the second is like unto it, Thou shalt love thy neighbour as thyself. On these two commandments hang all the law and the prophets. [KJV]

Many Christians believe that to the extent that one or another group of Christians do not subscribe to the Nicene Creed as being an accurate and complete summary of the Faith, they are not Christian. "Subscribe" as I use it here, does not necessarily mean that the Nicene Creed is recited in Worship.

The old creeds are useful summaries of Christian doctrine. They are useful because the way God decided to shape the Bible was not like a systematic theology textbook, even though we might sometimes wish it was!

The creeds were written in times of division in the church to clarify what the groups that wrote them believe the Bible taught. In general each phrase from one of the creeds will have several passages which back it up, though occasionally that is disputed, such as with the Filioque.

I think the real value of the creeds is that they were written to be completely unambiguous. If you talk to a non-Trinitarian about the passages which a Trinitarian believes support the Trinity, they will probably have alternative ways to interpret them: "it means one in person, the spirit isn't distinct from the father/son here, begotten indicates a created being". For better or worse, the creeds allow you to very quickly establish whether someone is an orthodox Christian or not. But to actually convince someone of orthodox doctrine you'll still need to go through those passages one by one.