  We describe the algebraic K-theory of the $K(1)$-local sphere and the
category of type 2 finite spectra in terms of K-theory of discrete rings and
topological cyclic homology. We find an infinite family of 2-torsion classes in
the $K_0$ of type 2 spectra at the prime 2, and explain how to construct
representatives of these $K_0$ classes.
