= Solution
For either normalized two-spinor $\chi^s$, $(\boldsymbol\sigma\cdot\mathbf p)^2=p^2I$ gives
$$
\psi_s^\dagger\psi_s=|N|^2\left(1+\frac{p^2}{(E+m)^2}\right)=|N|^2\frac{2E}{E+m},\qquad
\bar\psi_s\psi_s=|N|^2\frac{2m}{E+m}.
$$
The <Dirac equation> alone does not fix $N$: a normalization convention is needed. In the usual covariant external-spinor convention $\bar u_ru_s=2m\delta_{rs}$, choose real positive $N=\sqrt{E+m}$. Then $u_r^\dagger u_s=2E\delta_{rs}$ and $\bar u_s\gamma^\mu u_s=2p^\mu$. A second common convention is unit rest density, $\bar uu=1$, giving $N=\sqrt{(E+m)/(2m)}$ and density $E/m$. Both explain why the boosted density is not one particle per unit coordinate volume: the density is a <four-current> component and changes under a <Lorentz transformation>.
One may instead choose unit laboratory density, $N=\sqrt{(E+m)/(2E)}$, or box normalization $N=\sqrt{(E+m)/(2EV)}$ for one particle in volume $V$. An infinite plane wave is not normalizable to one particle over all space; a localized <wave packet> is required for that. Thus a unique $N$ cannot be inferred from the prompt without specifying which of these conventions is intended.
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