Fun fact: the King James Version of the Bible has several errors where it uses "thy" before a vowel sound. This might vary by edition, but I've verified some in a scan of the 1611 "he" bible. Unfortunately it's blackletter which I can't read quickly, and I really don't trust the OCR.

Semi-manual verse counts from a random digital copy I have convenient, before 'e' only:

  (2 thy, 2 thine) elder
  (2 thy, 0 thine) elect
  (19 thy, 3 thine) estimation (most in the same chapter!)
  ("thy ewe" is correct due to pronunciation)
  (1 thy, 0 thine) exceeding
  (1 thy, 1 thine) excellency
  (1 thy, 1 thine) expectation
  (2 thy, 110 thine) eye

2 John, being short, is the only book that exclusively does it wrong. The other errors are in Genesis, Exodus, Leviticus, Deuteronomy, Job, Proverbs, and Ezekiel, all of which also use the correct form.

(it also has errors before other vowels and 'h', though the 'u' one is debatable)

For "enemy" it is always the correct "thine".