The formation of a deeply-bound $K^- pp$ state with $I=1/2$, $J^pi=0^-$ by the $^3$He(in-flight $K^-$, $n$) reaction is theoretically investigated in a distorted-wave impulse approximation using the Greens function method. The expected inclusive and semi-exclusive spectra at $p_{K^-} = 1.0$ GeV/c and $theta_{rm lab} = 0^{circ}$ are calculated for the forthcoming J-PARC E15 experiment. We demonstrate these spectra with several types of phenomenological $K^-$-``$pp$ optical potentials $U^{rm opt}(E)$ which have an energy-dependent imaginary part multiplied by a phase space suppression factor, fitting to recent theoretical predictions or experimental candidates of the $K^-pp$ bound state. The results show that a cusp-like peak at the $pi Sigma N$ threshold is an unique signal for the $K^-pp$ bound state in the spectrum including the [$K^-pp$] $to$ $Y + N$ decay process from the two-nucleon $K^-$ absorption, as well as a distinct peak of the $K^-pp$ bound state. The shape of the spectrum is explained by a trajectory of a moving pole of the $K^-pp$ bound state in the complex energy plane. The importance of the spectrum with [$K^-pp$] $to$ $Y + N$ from the two-nucleon $K^-$ absorption is emphasized in order to extract clear evidence of the $K^-pp$ bound state.
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