How to find the temperature change of a solution?
The main problem here is a simple mistake in the algebra. You rearranged:  

$\pu{n\timesΔH_{sol} = m\times C\timesΔT}$  

to  

$\pu{ΔT = \frac{m\times C}{n\timesΔH_{sol}}}$  

rather than  

$\pu{ΔT = \frac{n\timesΔH_{sol}}{m \times C}}$  

Additionally, the molar enthalpy of solvation is given in the problem as units of kJ/g, not kJ/mol. Presumably, this is an error in the given problem. According to Parker, V . B ., Thermal Properties of Uni-Univalent Electrolytes, Natl . 
Stand . Ref . Data Series — Natl . Bur. Stand .(U.S.), No .2, 1965, the molar enthalpy of solution for $\ce{NH4Cl}$ is $\pu{14.78 kJ/mol}$.

There was also a slight miscalculation in the moles of solute. Where you calculated $\pu{0.165 moles}$ of $\ce{NH4Cl}$, you should have gotten:  

$\mathrm{8.5 g / 53.49\frac{g}{mol}=0.159~mol}$  

These errors corrected, plugging in the values to solve for $\Delta \text{T}$ is trivial and gives:  

$\pu{ΔT = \frac{\pu{0.159 mol}\times \pu{14.78 kJ mol-1}}{\pu{100g} \times \pu{4.186J~g-1~K-1}} = 5.6K}$