Multi-modal Learning has attracted widespread attention in medical image analysis. Using multi-modal data, whole slide images (WSIs) and clinical information, can improve the performance of deep learning models in the diagnosis of axillary lymph node metastasis. However, clinical information is not easy to collect in clinical practice due to privacy concerns, limited resources, lack of interoperability, etc. Although patient selection can ensure the training set to have multi-modal data for model development, missing modality of clinical information can appear during test. This normally leads to performance degradation, which limits the use of multi-modal models in the clinic. To alleviate this problem, we propose a bidirectional distillation framework consisting of a multi-modal branch and a single-modal branch. The single-modal branch acquires the complete multi-modal knowledge from the multi-modal branch, while the multi-modal learns the robust features of WSI from the single-modal. We conduct experiments on a public dataset of Lymph Node Metastasis in Early Breast Cancer to validate the method. Our approach not only achieves state-of-the-art performance with an AUC of 0.861 on the test set without missing data, but also yields an AUC of 0.842 when the rate of missing modality is 80\%. This shows the effectiveness of the approach in dealing with multi-modal data and missing modality. Such a model has the potential to improve treatment decision-making for early breast cancer patients who have axillary lymph node metastatic status.