We demonstrate that $(n,2n)$ reactions are induced by a high-flux pulse of fast neutrons ($\ensuremath{\sim}{10}^{10}$ neutrons in $\ensuremath{\sim}1\phantom{\rule{0.16em}{0ex}}\mathrm{ns}$) provided from a laser-driven neutron source (LDNS). The maximum energy of the broadband neutrons reaches a few tens MeV. Several kinds of metal targets are exposed to the fast neutrons. As a result, unstable isotopes, such as $^{54}\mathrm{Mn}, ^{58}\mathrm{Co}, ^{175}\mathrm{Hf}$, and $^{196}\mathrm{Au}$ are produced by $(n,2n)$ reactions and $^{180}\mathrm{Hf}^{m}, ^{181}\mathrm{Hf}, ^{56}\mathrm{Mn}, ^{198}\mathrm{Au}$, and $^{60}\mathrm{Co}$ are produced by $(n,\ensuremath{\gamma})$ reactions. We evaluate the neutron fluence and energy spectrum using the activation method in conjunction with a time-of-flight measurement. The neutron fluence is determined to be $(4.3\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{8}\phantom{\rule{0.16em}{0ex}}\mathrm{neutrons}/\mathrm{c}{\mathrm{m}}^{2}$ in the energy range from approximately 8 to 20 MeV at 8-mm downstream of the neutron source. The present scheme provides a method to evaluate high-density neutrons seen in stellar environments, which are expected to be generated from future LDNSs.