How do I root a piece of the above plant?

I found this info on About.com, hope it's what you are looking for. And to ATP!

"Fittonia propagates readily from leaf-tip cuttings. Take the leat-tip cutting in late spring or early summer, right around the same time you repot the plant. Make sure to include at least two growing nodes on the cutting to obtain the best results. Once you've potted up the cutting in a peat-based soil mix, you can expect fittonia cuttings to sprout roots within two to three weeks. Use of a rooting hormone is not usually necessary, but if your conditions are less than ideal (e.g., they are too dry or temperatures less than 80ºF on a continual basis), rooting hormone might increase your chances of success."

Wrap the cuttings in plastic and mist them. They like high humidity. I have read that you can root them in water and also can divide by roots.

Yes, I found that after, that they really like high humidity.