The vowel in question here is not standard in English. I had to learn it through careful imitation, but one day I realized it's not hard to explain. It's the same vowel as in "boot", except you don't round your lips. I wish I had been given this explanation!

(Note that this is not at all the same vowel as in "book".)

Bonus: If you've ever wondered about the mysterious Turkish dotless i (ı), this is also what that represents.

https://en.wikipedia.org/wiki/Close_back_unrounded_vowel