Impurity effects of Zn and Ni on the low-energy spin excitations were systematically studied in optimally doped La(1.85)Sr(0.15)Cu(1-y)A(y)O(4) (A=Zn,Ni) by neutron scattering. Impurity-free La1.85Sr0.15CuO4 shows a "spin gap" of omega=4 meV below T-c in the antiferromagnetic incommensurate spin excitation. In Zn:y=0.004, the spin excitation shows a spin gap of 3 meV below T-c. In Zn:y=0.008 and Zn:y=0.011, however, the dynamical susceptibility chi(") at omega=3 meV decreases below T-c and increases again at lower temperature, indicating an in-gap state. In Zn:y=0.017, the low-energy spin state remains unchanged with decreasing temperature, and elastic magnetic peaks appear below similar to 20 K and then exponentially grow. These results suggest that Zn induces a novel in-gap spin state, which becomes dominant and more static with increasing Zn. As for Ni:y=0.009 and Ni:y=0.018, the low-energy excitations below omega=3 meV and 2 meV disappear below T-c. The temperature dependence of chi(") at omega=3 meV, however, shows no upturn in contrast with Zn:y=0.008 and Zn:y=0.011, indicating the absence of an in-gap state. In Ni:y=0.029, though the magnetic signals were observed also at omega=0 meV, they showed no temperature dependence, implying that there is no static component. Furthermore, as omega increases, the magnetic peak width broadens and the peak position shifts toward the magnetic zone center (pi pi). We interpret the impurity effects as follows: Zn locally makes a nonsuperconducting island exhibiting the in-gap state in the superconducting sea with the spin gap. Zn reduces the superconducting volume fraction, thus suppressing T-c. On the other hand, Ni primarily affects the superconducting sea, and the spin excitations become more dispersive and broaden with increasing energy, which is recognized as a consequence of the reduction of energy scale of spin excitations. We believe that the reduction of energy scale is relevant to the suppression of T-c.