Nuclear level densities (NLDs) and $gamma$-ray strength functions ($gamma$SFs) have been extracted from particle-$gamma$ coincidences of the $^{92}$Zr($p,p gamma$)$^{92}$Zr and $^{92}$Zr($p,d gamma$)$^{91}$Zr reactions using the Oslo method. The new $^{91,92}$Zr $gamma$SF data, combined with photonuclear cross sections, cover the whole energy range from $E_{gamma} approx 1.5$~MeV up to the giant dipole resonance at $E_{gamma} approx 17$~MeV. The wide-range $gamma$SF data display structures at $E_{gamma} approx 9.5$~MeV, compatible with a superposition of the spin-flip $M1$ resonance and a pygmy $E1$ resonance. Furthermore, the $gamma$SF shows a minimum at $E_{gamma} approx 2-3$~MeV and an increase at lower $gamma$-ray energies. The experimentally constrained NLDs and $gamma$SFs are shown to reproduce known ($n, gamma$) and Maxwellian-averaged cross sections for $^{91,92}$Zr using the {sf TALYS} reaction code, thus serving as a benchmark for this indirect method of estimating ($n, gamma$) cross sections for Zr isotopes.
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