  This is the second in a series of two articles where we study various aspects
of the spectral theory associated to families of hyperbolic Riemann surfaces
obtained through elliptic degeneration. In the first article, we investigate
the asymptotics of the trace of the heat kernel both near zero and infinity and
we show the convergence of small eigenvalues and corresponding eigenfunctions.
Having obtained necessary bounds for the trace, this second article presents
the behavior of several spectral invariants. Some of these invariants, such as
the Selberg zeta function and the spectral counting functions associated to
small eigenvalues below 1/4, converge to their respective counterparts on the
limiting surface. Other spectral invariants, such as the spectral zeta function
and the logarithm of the determinant of the Laplacian diverge. In these latter
cases, we identify diverging terms and remove their contributions, thus
regularizing convergence of these spectral invariants. Our study is motivated
by a result from \cite{He 83}, which D. Hejhal attributes to A. Selberg,
proving spectral accumulation for the family of Hecke triangle groups. In this
article, we obtain a quantitative result to Selberg's remark.
