First derivation: the equation $Lm=H_2-H_1$ is only valid for a process at p=constant, which is not your case.
Second derivation: the equation $Q=C_p (T_2-T_1)$ is only valid for a process at p=constant, which is not your case. Plus the rest of the argument doesn't make sense: both $\Delta T=0$ and $\Delta V=0$ in a closed container with boiling water.