Firstly, the eqn: hot coin + cold water —> warm coin and warm water

heat gained by water = heat lost by coin= spht of coin J/gC * 13.0 g * (100-T)C

heat gained by water = 29.0 g H2O * 4.186 J/gC * (T-18.0)C

heat lost for Ni coin = 0.445 J/gC * 13.0 g * (100-T)C
heat lost for Ag coin = 0.233 J/gC * 13.0 g * (100-T)C

for Ni coin: 0.445 J/gC * 13.0 g * (100-T(Ni-final) )C = 29.0 g H2O * 4.186 J/gC * (T(Ni-final)-18.0)C

SOLVE for T(Ni-final)

T (Ag-final) ~ T (Ni-final) * (0.233 J/gC / 0.445 J/gC) = ??
check by solving for T(Ag-final) the way you did T(Ni-final)

Basic mathematics is a prerequisite to chemistry – I just try to help you with the methodology of solving the problem.