Solve for r instead of k^2
Clear[r, x]
Solve[{r/2 (x - Sin[x]) == 10, r/2 (1 - Cos[x]) == 2}, {r, x}, Reals] // N
{{r -> 3.57932, x -> 4.59459}, {r -> 2.66039, x -> 8.38167}, {r -> 2.02561, x -> 9.65013}}
Now take the square root of each r to get k. Mathematica having hard time with the k^2 being there, even for positive k
(notice, you write sin and cos. In Mathematica, these should be Sin and Cos
