$(1)$
\[\begin{align}
Q_1=mc\Delta T
&=5.0×4.2×(100-20)\\&=1680\,\rm{kJ}
\end{align}
\]
\[
\begin{align}
Q_2=mr
&=5.0×2257\\&=11285\,\rm{kJ}
\end{align}
\]
\[
\therefore Q=Q_1+Q_2=12965\,\rm{kJ}
\]
$(2)$
\[\begin{align}
t=\frac{Q}{P}=\frac{12965\,\rm{kJ}}{0.8\,\rm{kW}}&=16206.25\,\rm{s}\\
&=270\,\rm{min}\\
&=4\,\rm{h}30\,\rm{m}
\end{align}\]
