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

Americans pronounce kept as KEHP (/kɛp/). You'll hear it in sentences like "She kept her promise to help" or "He kept quiet during the entire film" — more examples below.
Record yourself saying "kept" 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.

Drop your jaw moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

Press your lips together to stop the air, then release. No vocal cord vibration.

Click any sentence to see the full breakdown — every link, every reduction, every flap-T.
The textbook way isn't wrong — it's just not how anyone actually says it.
In "kept", the "p" 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.