= Solution
Treat the ions as a stationary neutralizing background and displace the cold electrons by a small longitudinal distance $\xi$. Their <electric polarization> is $P=-n_ee\xi$, so the restoring electric field is $E=-P/\epsilon_0=n_ee\xi/\epsilon_0$. The electron equation of motion is
$$
m_e\ddot\xi=-eE=-\frac{n_ee^2}{\epsilon_0}\xi,\qquad
\boxed{\omega_{\rm pe}=\sqrt{\frac{n_ee^2}{m_e\epsilon_0}}.}
$$
For $E=E_0\cos\omega_{\rm pe}t$, the oscillatory velocity is $v=-eE_0\sin\omega_{\rm pe}t/(m_e\omega_{\rm pe})$. Hence
$$
\boxed{\langle u_K\rangle=\frac12n_em_e\langle v^2\rangle=\frac{\epsilon_0E_0^2}{4}=\left\langle\frac{\epsilon_0E^2}{2}\right\rangle.}
$$
The longitudinal mode exchanges electron kinetic energy and electrostatic field energy like a <harmonic oscillator>, whose mean kinetic and potential energies are equal over a cycle. This is a time-average statement for the coherent oscillation, not thermal equipartition with $k_{\rm B}T$.
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