Stuart at Trail Spotting recommends the Kauai – Alaka’i Swamp Trail 11mi (17.7km). Hard.
… Take a westerly drive around Kauai until you run out of road and you’ll find yourself overlooking the plunging cliffs of Kalalau Valley on the fabled Na Pali Coast.
The only way to explore beyond this point by land is on foot, and the well-maintained Pihea trail leading to the Alaka‘i Swamp trail is an exhilarating way to do so. Hike along cliff edges, through jungle and across misty mountain-top swamps to reach the trail’s end and clifftop views over Wainiha Pali. …
Click through for photos and maps: Trail Spotting
besthike.com 
We did three hikes in Kauai, and if you were looking at a day-hike recommendation then it would be this one without hesitation.
Unless, that is, you think you could tackle the fabled Kalalau trail along the Na Pali coast, which is 11 miles each way and tackled only as a backpacking trip unless you’re feeling superhuman.
Aside from following the trail for two miles to get to Hanakapi’ai Falls, we haven’t tackled the Kalalau trail yet. Also, if you’re not familiar with this Na Pali coast trail I recommend reading about some of the folk who have hiked its length.