We introduce BIONIC (duraBle tImekeeper fOr iNtermIttent Computing), a novel power system architecture and software runtime that facilitates time-sensitive intermittent computing for battery-less Internet of Things. BIONIC integrates timekeeping into energy storage hardware: its power system comprises two energy storage capacitors. BIONIC switches between these two energy storage capacitors to estimate ambient power. Using these estimations, BIONIC can predict and track charging times (i.e., off-time durations) to schedule time-sensitive tasks and complete them on time. Our evaluations showed that BIONIC significantly extends the measurable off-time intervals by a factor of 15 to 1620 compared to state-of-the-art, while achieving an accuracy of up to 99.1% and effectively adapting to new energy conditions. Besides, in a typical batteryless application, BIONIC's power-aware intermittent computing runtime boosted the number of completed time-sensitive operations by 30% while reducing failed timely operations by 29%.