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

Americans pronounce contentedly as kuhn-TEHN-tuhd-lee (/kənˈtɛntədli/). In "contentedly", the "t" right after N is dropped — the tongue skips the T stop and moves directly from the N position to the next sound. This is called the Silent T after N, a hallmark of natural-sounding American speech. It comes out as kuhn·TEHN·tuhd·lee. Stress falls on the second syllable — keep everything else short and quick. You'll hear it in sentences like "The cat purred contentedly while sitting on her lap".
Record yourself saying "contentedly" and play it back. The mic stays on your device — nothing's uploaded.
4 syllables, 11 sounds. Tap a syllable to jump to its row, then 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.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

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.

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

Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

The T is skipped entirely. Your tongue doesn't make contact at the T position.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

Place the tip of your tongue against the alveolar ridge just behind your top front teeth, the same contact point as /t/, /d/, and /n/. The difference is what happens to the air: for /l/, you let it flow continuously around the <em>sides</em> of the tongue (that's why /l/ is called a lateral). Turn your voice on the whole time. Lips stay relaxed, no rounding or flaring. For the Dark L variant at the end of a syllable, also pull the back of the tongue up and back toward the soft palate.

Pull the corners of your lips back slightly. Arch the middle-front of your tongue high toward the roof of the mouth.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "contentedly", the "t" right after N is dropped — the tongue skips the T stop and moves directly from the N position to the next sound. /t/ is completely silent — the tongue skips the T stop and moves directly from the N position to the next sound.
In "contentedly", the "d" 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.
Stress falls on the second syllable, not the others. Stretch TEHN — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.