This paper proposes a robust digital signal recovery (DSR) technique to tackle the high signal-to-noise ratio (SNR) variation and transmitter memory effects for broadband power efficient down-link in next-generation low Earth orbit (LEO) satellite constellations. The robustness against low SNR is achieved by concurrently integrating magnitude normalization and noise feature filtering using a filtering block built with one batch normalization (BN) layer and two bidirectional long short-term memory (BiLSTM) layers. Moreover, unlike existing deep neural network-based DSR techniques (DNN-DSR), which failed to effectively take into account the memory effects of radio-frequency power amplifiers (RF-PAs) in the model design, the proposed BiLSTM-DSR technique can extracts the sequential characteristics of the adjacent in-phase (I) and quadrature (Q) samples, and hence can obtain superior memory effects compensation compared with the DNN-DSR technique. Experimental validation results of the proposed BiLSTM-DSR with a 100 MHz bandwidth OFDM signal demonstrate an excellent performance of 11.83 dB and 9.4% improvement for adjacent channel power ratio (ACPR) and error vector magnitude (EVM), respectively. BiLSTM-DSR also outperforms the existing DNN-DSR technique in terms of the ACPR and EVM by 2.4 dB and 0.9%, which provides a promising solution for developing deep learning-assisted receivers for high-throughput LEO satellite networks.