Accessing high-quality video content can be challenging due to insufficient and unstable network bandwidth. Recent advances in neural enhancement have shown promising results in improving the quality of degraded videos through deep learning. Neural-Enhanced Streaming (NES) incorporates this new approach into video streaming, allowing users to download low-quality video segments and then enhance them to obtain high-quality content without violating the playback of the video stream. We introduce BONES, an NES control algorithm that jointly manages the network and computational resources to maximize the quality of experience (QoE) of the user. BONES formulates NES as a Lyapunov optimization problem and solves it in an online manner with near-optimal performance, making it the first NES algorithm to provide a theoretical performance guarantee. Comprehensive experimental results indicate that BONES increases QoE by 5\% to 20\% over state-of-the-art algorithms with minimal overhead. Our code is available at https://github.com/UMass-LIDS/bones.