Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Americans pronounce kite as KAHYT (/kaɪt/). You'll hear it in sentences like "Keep the kite" or "The size of the fine kite was quite high" — more examples below.
Record yourself saying "kite" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. Explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Start with your jaw open wide and your tongue resting low and flat. Glide the front of your tongue up toward the roof of your mouth as your jaw closes halfway.
Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "kite", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.