What should you charge for a lesson?

Put in the income you want from teaching and the weeks you actually teach. The sum is the one in the guide: divide by the teaching weeks, not by 52. Nothing is stored and there is nothing to sign up for.

This is arithmetic, not a recommendation: it tells you what your own target implies, not what the market pays. For where to sanity-check it, and how to put your rates up without losing anybody, read How much to charge for music lessons.