The characteristics of Fe-based superconductors are manifested in their electronic, magnetic properties, and pairing symmetry of the Cooper pair, but the latter remain to be explored. Usually in these materials, superconductivity coexists and competes with magnetic order, giving unconventional pairing mechanisms. We report on the results of the bulk magnetization measurements in the superconducting state and the low-temperature specific heat down to 0.4 K for ${\text{BaFe}}_{2$-${}x}{\text{Ni}}_{x}{\text{As}}_{2}$ single crystals. The electronic specific heat displays a pronounced anomaly at the superconducting transition temperature and a small residual part at low temperatures in the superconducting state. The normal-state Sommerfeld coefficient increases with Ni doping for $x=0.092$, 0.096, and 0.10, which illustrates the competition between magnetism and superconductivity. Our analysis of the temperature dependence of the superconducting-state specific heat and the London penetration depth provides strong evidence for a two-band $s$-wave order parameter. Further, the data of the London penetration depth calculated from the lower critical field follow an exponential temperature dependence, characteristic of a fully gapped superconductor. These observations clearly show that the superconducting gap in the nearly optimally doped compounds is nodeless.