It may be easy to see how the Gospels are "inspired," but what about the rest of the New Testament?
Article VI of the XXXIX Articles states:
VI. Of the Sufficiency of the Holy Scriptures for Salvation.
Holy Scripture containeth all things necessary to salvation: so that whatsoever is not read therein, nor may be proved thereby, is not to be required of any man, that it should be believed as an article of the Faith, or be thought requisite or necessary to salvation. In the name of the Holy Scripture we do understand those canonical Books of the Old and New Testament, of whose authority was never any doubt in the Church. ... All the Books of the New Testament, as they are commonly received, we do receive, and account them Canonical.
So what's the rationale for this?
Apostolic authority, for one thing. Jesus promised that the Spirit would bring all things to the disciples's remembrance.
John 14:25,26 25 “These things I have spoken to you while being present with you. 26 But the Helper, the Holy Spirit, whom the Father will send in My name, He will teach you all things, and bring to your remembrance all things that I said to you.
John 16:12-13 also says:
12 “I still have many things to say to you, but you cannot bear them now. [Jesus promises more instruction to the future apostles]
13 However, when He, the Spirit of truth, has come, He will guide you into all truth; [the Spirit is going to guide them into all the true things they need to know, believe and pass on to the rest of the Church]
for He will not speak on His own authority, but whatever He hears He will speak;
[One of the main reasons for the coming of the Holy Spirit was to do this task: continuing the teaching of Jesus]
and He will tell you things to come.
[compare this with Revelation 1:10 - John ties the revelation of Jesus to the presence of the Holy Spirit in his soul]
Paul speaks of this very action of the Spirit in his own teaching: I Cor. 2:12,13 12 Now we have received, not the spirit of the world, but the Spirit who is from God, that we might know the things that have been freely given to us by God. 13 These things we also speak, not in words which man’s wisdom teaches but which the Holy Spirit teaches, comparing spiritual things with spiritual.
See also Ephesians 3: 3 For this reason I, Paul, the prisoner of Christ Jesus for you Gentiles— 2 if indeed you have heard of the dispensation of the grace of God which was given to me for you, 3 how that by revelation He made known to me the mystery (as I have briefly written already, 4 by which, when you read, you may understand my knowledge in the mystery of Christ), 5 which in other ages was not made known to the sons of men, as it has now been revealed by the Spirit to His holy apostles and prophets: [note also the use of the plural]
Note how Paul describes his own teaching: I Thessalonians 2:13. 13 For this reason we also thank God without ceasing, because when you received the word of God which you heard from us, you welcomed it not as the word of men, but as it is in truth, the word of God, which also effectively works in you who believe.
It is no wonder that the apostle Peter equates Paul's writings to the Old Testament scriptures: 2 Peter 3: 14 Therefore, beloved, looking forward to these things, be diligent to be found by Him in peace, without spot and blameless; 15 and consider that the longsuffering of our Lord is salvation—as also our beloved brother Paul, according to the wisdom given to him, has written to you, 16 as also in all his epistles, speaking in them of these things, in which are some things hard to understand, which untaught and unstable people twist to their own destruction, as they do also the rest of the Scriptures.
The Acts of the Apostles is only Part Two of one work by Luke. It is taken as having the same authority as Part One, the Gospel.
So much for the rationale. There are also instructions in the NT of how the epistles are to be used that show they are to be taken as authoritative (e.g. 1 Thess. 5:27: 27 I charge you by the Lord that this epistle be read to all the holy brethren. 2 Thess. 15 Therefore, brethren, stand fast and hold the traditions which you were taught, whether by word or our epistle.)