The nut on the LU21C is glued to the fret board, not tapped in. I was afraid that if I tried removing it, I would damage the finish.
After doing a lot of research and actually avoiding the Lanikai for a few days (because the $30 Mahalo was easier to play), I decided on a plan of action.
Besides the discomfort of the high action, it caused the first fret notes to play very sharp. I understood why - the strings were stretching more than they should, to get from the high nut (0 fret) to the fingerboard behind the first fret. I decided if the action were too low, the fretted note would be flat, and fretting higher on the fingerboard would cause fret buzz, as the oscillation of the string would bring it into contact with additional frets.
With these things in mind, I bought a set of acetylene torch tip cleaners from Bunnings (like Lowes for those in the US) for 7.95. For each string, I chose the size that most closely matched the string width. I followed these steps:
1. Loosen the string enough to ease it out of the nut slot.
2. Gently deepen the nut slot a small amount, being careful not to change it's shape or angle.
3. Re-tune the string and check the first fret.
4. Check for fret buzz by plucking the string very hard at each fret.
I kept doing this until the first fret was just a few cents sharp. I wanted to be conservative and not file off too much.
I ended up having to file off quite a bit more than I expected. I have no fret buzz on any fret, and the first fret is now in tune, at least good enough for my ears. My tuner says it is about 1 or 2 cents sharp. Action is noticeably lower, and the first fret only takes a little more force than the other frets, to get a clean note. I have made the nut slots deep enough that I need to sand off the height of the nut if the strings are to sit 1/3 above the nut. But it does not seem to affect the sound, or I am not sophisticated enough to hear it.
I would not recommend doing this if you do not have a luthier or guitar shop to fall back on! I knew that if I screwed it up I would just take it in and have a new nut fitted.