Physical layer secret key generation (PLKG) is a promising technology for establishing effective secret keys. Current works for PLKG mostly study key generation schemes in ideal communication environments with little or even no signal interference. In terms of this issue, exploiting the reconfigurable intelligent reflecting surface (IRS) to assist PLKG has caused an increasing interest. Most IRS-assisted PLKG schemes focus on the single-input-single-output (SISO), which is limited in future communications with multi-input-multi-output (MIMO). However, MIMO could bring a serious overhead of channel reciprocity extraction. To fill the gap, this paper proposes a novel low-overhead IRS-assisted PLKG scheme with deep learning in the MIMO communications environments. We first combine the direct channel and the reflecting channel established by the IRS to construct the channel response function, and we propose a theoretically optimal interaction matrix to approach the optimal achievable rate. Then we design a channel reciprocity-learning neural network with an IRS introduced (IRS-CRNet), which is exploited to extract the channel reciprocity in time division duplexing (TDD) systems. Moreover, a PLKG scheme based on the IRS-CRNet is proposed. Final simulation results verify the performance of the PLKG scheme based on the IRS-CRNet in terms of key generation rate, key error rate and randomness.