The use of italics in Bible translation predates the KJV and is a complex topic. A good introduction is here:
Some excerpts:
The original KJV of 1611, like its predecessor, the Bishops' Bible,
was printed in black-letter type. Use was again made of roman type for
words supplied by the revisers, but not found in the original
languages. Numerous changes have been made in subsequent editions of
the KJV. A few were unintentional, but most were deliberate attempts
to correct errors. l1 IQ 1612 an edition in octavo was printed using a
small clear roman type, and introducing the use of italics in this
version. This was followed by a similar edition in 1616 also in roman
type. The 1762 revision by Thomas Paris, published at Cambridge,
extended and improved in accuracy the use of the italics. In 1769 the
Oxford edition by Benjamin Blayney made more corrections and further
extended the use of italics, probably beyond the limits that the
original famous 47 revisers would have approved.
Reasons for italics include:
Sometimes there was a deviation between TR and another text, and if the translators believed the other text was correct, they would add the word in italics.
When there were explanatory words useful for understanding the text but not present, italics could be used.
- consistency with previous versions
The KJV was not a new translation but an update of existing well-regarded translations such as the Bishop's Bible and Geneva Bible, and when there was deviations with previous famous translations, italics might be used, or when the previous versions had italics, these might be carried over into the KJV.
Also keep in mind that the italics were revised over time, so you would need to check in which version of the KJV the italics appear, and then compare to the Hebrew Source text, the First Rabbinic Bible or Biblia Rabbinica, edited by Felix Pratensis, a jew who converted to Christianity, in 1517. Then look at other older Bible translations in order to try track down the reason for the italics.
Unfortunately I don't have access to the First Rabbinic Bible -- perhaps someone else on this site does -- so I can't confirm whether it has a missing yod in Deut 8.17 which might cause the KJV translators to add "mine" with italics. Or it could have been something as simple as a smudge that made the yod uncertain in the copy of the the rabbinic bible they had. That, or simply a printer's error, would explain the italics. In the Geneva Bible it is not italicized.