Measurements of B-s2*(5840)(0) and B-s1(5830)(0) mesons are performed using a data sample of proton-proton collisions corresponding to an integrated luminosity of 19.6 fb(-1), collected with the CMS detector at the LHC at a centre-of-mass energy of 8 TeV. The analysis studies P-wave B-s(0) meson decays into B(*)(+)K(-)and B(*)K-0(S)0, where the B+ and B-0 mesons are identified using the decays B+ -> J/psi K+ and B-0 -> J/psi K* (892)(0). The masses of the P-wave B-s(0) meson states are measured and the natural width of the B-s2*(5840)(0) state is determined. The first measurement of the mass difference between the charged and neutral B* mesons is also presented. The B-s2*(5840)(0) decay to (BKS0)-K-0 is observed, together with a measurement of its branching fraction relative to the B-s2*(5840)(0) -> B+K- decay.