If the Bible contains all of the word of God and there can be no more, then why do we need the Nicene Creed? There seems to be a huge contradiction in traditional Christianity that claims that there can be no more scripture or divine doctrine and yet the only way to determine whether a denomination is "Christian" (e.g., Jehovah's Witnesses or The Church of Jesus Christ of Latter-day Saints) is to refer to an extra-Biblical doctrine (i.e., the Nicene Creed) as if it were scripture.

(If you don't think the Nicene Creed is treated as scripture, then how do you define scripture?)
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.