To what extent can we be sure that the red letters in a KJV Bible are the actual words of Jesus?
The 'red letters' are not themselves part of the KJV translation. The red letters appear in many different translation of the Bible.
To answer the question: no, there is not universal agreement about exactly what words in the Bible were spoken by Jesus. The New Testament languages did not include punctuation like quote marks. Most of the time it is pretty clear from the text where Jesus words begin and end. Occasionally however it is hard to tell where a quote of Jesus ends and where an explanation from the Gospel author begins. For example in John 3 there is some uncertainty about whether Jesus quote which begins in verse 10 continues to verse 21 or stops after verse 15, with the remainder being explanation by the author.
If you are asking about actual words, then Jesus probably spoke in Aramaic, and so even the original Greek is not an untranslated quote. English and other modern language renditions are further translations.
TL;DR to the same extent we can be sure that the KJV translation is accurate at all
Bible translators use their understanding of ancient Greek, Hebrew, and Aramaic to study the different Biblical manuscripts that they have available to them to determine how to translate the words. Since the manuscripts don't use quotation marks, the translation process includes determining which words are actually Christ's words. The practice of highlighting Christ's words in red text was started by Louis Klopsch in 1899. It's especially helpful in the KJV and in other translations that don't use quotation marks.
It's also important to know that not all of the words highlighted in red are where the author quoted Christ. Some Bibles also highlight Christophanies (basically visions of Christ) in the Old Testament and the Epistles.
Whether the words are red or black, those are the words of Jesus. Of course the words were originally written in Greek, Latin, or Hebrew. But nevertheless, these are the very words of Jesus through the eyewitnesses of those people in that time. Some translations are questionable.(I.E. The Message by Eugene Peterson) Some versions of the Bible translate it either word for word, or thought for thought. But overall, they are Inspired by God and written by Godly men.
The Red Letter KJV shows the words attributed to Jesus in red. Translation differences aside, it is the strong consensus of scholars that we do not know the actual words Jesus used. Rex Weyler, in The Jesus Sayings, page 14, cites Robert J. Miller, editor of The Complete Gospels: "Some of the words attributed to Jesus were not actually spoken by him. While this is not news to scholars, it is news to the ... public."
The Jesus Seminar, a theological research group, sought a consensus among theologians and scholars as to whether sayings attributed to Jesus were likely to have been spoken by him at all. It used voting of members as criteria for establishing what Jesus said, with weighted member votes:
3 - Jesus did say the passage quoted, or something very much like it; 2 - Jesus probably said something like the passage; 1 - Jesus did not say the passage, but it contains Jesus' ideas; 0 - Jesus did not say the passage.
The criteria for acceptance were based on orality, irony and whether Jesus was telling his listeners to trust in God. The criteria for non-acceptance were based on self-reference, whether the words framed or introduced other material, dealt with Christian issues, or represented a theological agenda specific to one gospel.
Even by voting '3' - Jesus did say the passage quoted, or something very much like it - the participants were not claiming that his actual words were somehow remembered and written down years later. The Seminar concluded that in the four canonical gospels and the Gospel of Thomas, only about 18 per cent of sayings attributed to Jesus were likely to have been said by Jesus himself, with nearly all the sayings in John's Gospel seen as inauthentic.