Two other references to similar sums are
Bruce C. Berndt and Boon Pin Yeap, Explicit evaluations and reciprocity theorems for finite trigonometric sums, Advances in Applied Mathematics Volume 29, Issue 3, October 2002, Pages 358--385
and
Ira Gessel, Generating Functions and Generalized Dedekind Sums, Electronic J. Combinatorics, Volume 4, Issue 2 (1997) (The Wilf Festschrift volume), R11.
The paper of Berndt and Yeap uses contour integration and has an extensive list of references. My paper uses elementary methods, including partial fractions.