E-ink displays are bi-stable and only consume microwatts when refreshed. Even optimized, the processor, wifi and front light consumption will be significant.
Accurate figures are not easy to pin down. With many possible variables, actual power consumption is best
measured. Display manufacturer's claim that it takes about 7 mj/cm2 for a full screen switch (refresh). If a refresh is < 400 ms, then a full screen refresh would use about .014 mAh per refresh or some 20 mAh per day. Which would be about 77 days per charge if we were to ignore processor and wifi power.
Real life figures would probably see screen updates using far less current. But dwarfed by processor and wifi current. As a clock the Kindle should be able to spend the majority of time suspended so it would seem reasonable to expect an optimized clock to draw less power than regular reading. So > 6 weeks if you believe Amazon's battery claims for the PW4.