We examined nearest-neighbor spacing (NNS) statistics in doubly excited states of helium near the double ionization threshold. Using the Brody parameter $q$ to measure the NNS distribution between the regular Poisson distribution $(q=0)$ and the chaotic Wigner distribution $(q=1)$, we showed that for levels near the $N=20$ threshold of ${\mathrm{He}}^{+}$, or at about $0.13\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ below the double ionization threshold, the NNS distribution has $q=0.66$. The result shows the slow approach of the NNS of helium energy levels towards the Wigner distribution vs the excitation energy. Using an $s$-wave model where the angular momentum of each electron is restricted to zero, we also examined the NNS for levels up to the $N=30$ threshold of ${\mathrm{He}}^{+}$. We showed the gradual increase in $q$ as the excitation energy is increased. To generate the theoretical data needed for the NNS analysis, we have used the hyperspherical close-coupling method, with the recently proposed diabatization of potential curves and the truncation of channels, to greatly reduce the complexity in the calculation. We also investigated the dependence of $q$ vs the nuclear charge in the same scaled energy region, and for different symmetries, to assess their relation with the rate of approaching the $q=1$ Wigner distribution.