The efficiency of link adaptation in wireless communications relies greatly on the accuracy of channel knowledge and transmission mode selection. In this paper, a novel deep learning based link adaptation framework is proposed for the orthogonal frequency-division multiplexing (OFDM) systems with compressed-sensing-assisted index modulation, termed as OFDM-CSIM, communicating over millimeter-wave (mmWave) channels. To achieve link adaptation, a novel multi-layer sparse Bayesian learning (SBL) algorithm is proposed for accurately and instantaneously providing the required channel state information. Meanwhile, a deep neural networks (DNN)-assisted adaptive modulation algorithm is proposed to choose the best possible transmission mode to maximize the achievable throughput. Simulation results show that the proposed multi-layer SBL algorithm enables more accurate channel estimation than the conventional techniques. The DNN-based adaptive modulator is capable of achieving a higher throughput than the learning-assisted solution based on the k nearest neighbor (k-NN) algorithm, and also the classic average signal-to-noise ratio (SNR)-based solutions. Moreover, analysis shows that both the multi-layer SBL algorithm and the DNN-assisted adaptive modulator achieve better performance than their respective conventional counterparts while at a significantly lower computational complexity cost.