12 days? I love Hawaii and that's a long time on Kauai. Been goign there for years, last time was about 3 or 4 years ago. Not much has changed. If you've been, you know how slow it can get there, but that's the appeal.
A good local guide is this book: http://www.explorekauai.com/.
Lots of off the beaten path things. Had some good recs for secluded beaches up near Princeville.
If you're a golfer, Princeville is a must. I try to for twilight hours just when it starts. You might get lucky if you get there early. They let me tee off about an hour early and I was able to finish. Poipu has a great course as well. The local course near Kapa'a is cheap and much better than a lot of muni courses on the mainland. Heck, you're in Hawaii...
Hamura's Saimin is a must for local Saimin. Great lilikoi (passionfruit) pie. For FANTASTIC (and I mean that) fried chicken, there's a hole in the wall near Hanapepe called Imu Hut. Good to stop by, get a bucket of chicken for picnic and head up to Waimea Canyon. For a great breakfast, try the Country Kitchen. I loved the macadamia pancakes with coconut syrup.
For some reason, there's a big concentration of hippy/health nuts there too. Worked out well though, my wife made some connections at the local yoga studio and we were able to find some babysitters. We had a good vegan meal at Lotus (?). Across the street there's an organic burrito joint too.
As I think of more, I'll post. Bear in mind that this is a SMALL island, Maui (Lahaina) is New York City compared to Kauai. Stay off the main road through town (Kihei) during lunch and afternoon commute. You'll be in traffic for an hour.