The Body of Believers, the Early Church, through an organic process of regular use of the various books of the NT as divine scripture, established the canon of the NT. The councils of Hippo Regius (393 AD) and Carthage (397 AD) merely formally-acknowledged what the Body of Believers had already settled upon as the canon of the NT. The western Church arrived at the 27 books of the NT earlier than the eastern Church, but by 508 AD, both western and eastern branches of the Church were agreed upon the 27 books being canonical.
The OT canon was established in much the same way as the NT, the Hebrew council of Jamnia (90 AD) formally acknowledging what the Jews had, through prolonged and constant use, already settled upon as the OT canon and had informally closed to additions by 200 BC.
How is the Bible now our sole authority for Christian belief and practice? Well, for me, it's a matter of respecting what I believe was a Spirit-guided process through which the Early Church settled upon the canon of the Bible. There were many letters the apostles wrote, but only a select few were received by the Early Church as authoritative scripture. I am content to follow their lead, accepting what they agreed was, through prolonged and common use of it as such, the divinely-inspired canon of the Bible. By 400 AD, this process had well-established the canon - and closed it to further additions.
The NT canon was written by men acknowledged by the Early Church as apostles, not merely disciples, of Jesus Christ, who had been charged by him directly with the establishment and teaching of the Church, imbued by God with supernatural power in manifestation of their special spiritual authority. They had unique right as such to dictate the doctrines and practices of the Church, which the Church acknowledged by its general submission and conformity to their teaching and authority (though, there were rebellious false teachers almost from the start of the Church). There have been no other men since the time of these apostles who have had a legitimate claim to their authority, having never received from Christ in-person the charge to serve as his apostle to the Church, nor performed the same miraculous, authority-establishing feats with the regularity, and to the degree, of the first apostles, nor having the general acknowledgment of the Church universal as such, as did the first apostles.
Since there have not been any other apostles legitimately issuing truth from God to the Church, and since the canon of Scripture has long been closed, I rely on the Bible as it is as the Final Word from God on all Christian belief and practice - just as the Early Church had settled into doing within the first 300 years of its existence.
Is the Bible the only source of Truth? On spiritual matters, yes. But there are regions of Truth about which the Bible has little to say. We find nothing about algebra, or chemistry, or medicine, or space-flight engineering in the Bible. There is plenty of truth in these fields of knowledge, however.
Matthew 18:17 (NASB)
17 "If he refuses to listen to them, tell it to the church; and if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector.
At the time Christ spoke these words, there was no Roman Catholic Church, no pontiff, or archbishops, or marriage-abstinent clergy, no papal bulls, or indulgences, or Inquisitors. The "Church" was the common community of believers, not the ugly, institutionalized, power-mongering monstrosity that is the Roman Catholic Church of today (and history).
Matthew 5:14 (NASB)
14 "You are the light of the world. A city set on a hill cannot be hidden;
As all disciples of Christ are, the apostles were "the light of the world," yes, but they are long gone, leaving only the canon of the NT behind. There have been no other apostles of their sort since they died off - not even if they are wearing garish robes and a big, silly hat.
1 Timothy 3:13-15 (NASB)
13 For those who have served well as deacons obtain for themselves a high standing and great confidence in the faith that is in Christ Jesus.
14 I am writing these things to you, hoping to come to you before long;
15 but in case I am delayed, I write so that you will know how one ought to conduct himself in the household of God, which is the church of the living God, the pillar and support of the truth.
The Church is the "pillar and support of the truth," which is God's Truth given to us in His word, the Bible. The Church, Paul wrote here, supports (or upholds) the truth; it doesn't produce God's truth.