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Computer-aided pathology diagnosis based on the classification of Whole Slide Image (WSI) plays an important role in clinical practice, and it is often formulated as a weakly-supervised Multiple Instance Learning (MIL) problem. Existing methods solve this problem from either a bag classification or an instance classification perspective. In this paper, we propose an end-to-end weakly supervised knowledge distillation framework (WENO) for WSI classification, which integrates a bag classifier and an instance classifier in a knowledge distillation framework to mutually improve the performance of both classifiers. Specifically, an attention-based bag classifier is used as the teacher network, which is trained with weak bag labels, and an instance classifier is used as the student network, which is trained using the normalized attention scores obtained from the teacher network as soft pseudo labels for the instances in positive bags. An instance feature extractor is shared between the teacher and the student to further enhance the knowledge exchange between them. In addition, we propose a hard positive instance mining strategy based on the output of the student network to force the teacher network to keep mining hard positive instances. WENO is a plug-and-play framework that can be easily applied to any existing attention-based bag classification methods. Extensive experiments on five datasets demonstrate the efficiency of WENO. Code is available at https://github.com/miccaiif/WENO.