Eye tracking has been increasingly recognised as a valuable tool for assessing cognitive health. However, many existing eye tracking systems are often expensive, imprecise, and inconvenient, which limits their accessibility and effectiveness in clinical settings. To address these challenges, we propose a novel deep learning-based method for pupil centre and eye corner localisation using a single camera, integrating a Nested UNet, Gaussian heatmaps, Top-K averaging, and Adaptive Wing Loss. We also present a bespoke proof-of-concept hardware system designed to be compatible with our eye-tracking method, making it well-suited for conveniently administering eye-tracking tests for cognitive health monitoring. Our eye tracking model demonstrated superior localisation results on the benchmark GI4E dataset, when compared with other state-of-the-art methods, by achieving an accuracy of 99.11% with a tolerance of 0.025 normalised error. Additionally, preliminary qualitative results exhibited outstanding generalisation performance on a dataset captured by the proof-of-concept hardware system. The proposed eye tracking model and hardware system will be used in future work to develop accessible, user-friendly, and cost-effective solutions for cognitive health monitoring and disease diagnosis.