What you have is Colocasia esculenta. They are surprisingly hardy and may even survive a light frost. Wait until a hard frost kills the leaves, then cut the leaves off, dig up the tubers and let them dry out. In the meantime, stop fertilizing and cut back on watering to encourage them to go dormant.

I generally cut off the roots and leaves, lay the tubers on newspapers and let them air-dry for a few days, rotating them occasionally so they dry out evenly. As they dry out you can peel the fleshy leaf bases off the tubers. You will find one big tuber at the base of the leaves, and usually several smaller tubers around it that you can break off and dry separately. Store them cool and dry. I store them in paper bags in my cold attic over the winter. I usually lose one or two to rot anyway.