The chemotaxis sensory system enables Escherichia coli to detect and migrate in chemical gradients of nutrients and toxins. To do that this bacterium processes information on different time scales, from fast signalling by receptors to slow directional changes of the motors. Additionally, adaptation by covalent receptor modification remembers chemical concentrations from seconds ago. To account for the history dependence of these processes we focus on single-cell trajectories, and integrate tracking of cells in a microfluidic device, information theory, and simulations of swimming bacteria. We aim to relate performance of chemotaxis as measured by the drift velocity up the gradient, with information-processing ability and its thermodynamic cost. A new design principle may emerge: maximising information transmission maximises performance.