I think perhaps a market survey to estimate the demand for your product is warranted before you invest much more time in your project.
I used to shave every day, and as
@KrisBlueNZ said, hair became stuck in my razor. Sometimes I just swished it out, sometimes I just changed blades, and finally I just stopped shaving. Phil Roberts, of Duck Dynasty fame, observes that only women and young males do not grow beards. It is a most natural thing for a man to do. When I was a young male, I looked forward to the day when I too could take cold steel and scrape it across my face in a morning ritual of manhood. This absurd practice continued for many years as I tried many kinds of razors, including electric shavers, in attempts to achieve a smooth face. Some practices were more successful than others, but all were time consuming and, ultimately, I decided, unnecessary. I quit shaving quite a few years ago, although I do from time to time trim the ends of my beard a bit. My current wife loves my soft, silky beard. I will not be a customer for your ultraviolet light-sanitizing, water-jet, beard-hair extraction system. You might consider whether a Water Pic tooth blaster would serve the same purpose, sans sanitation.
I think I do remember reading that a germicidal UV light-emitting diode is now available. So your idea is practical, could be made quite small, and could operate from a pair of AA dry cells for awhile at least. My Water Pic was battery operated but totally sealed, being rechargeable through an inductance-coupled stand. I never worried about "sanitizing" the water-jet tips though. Just filled the tank with Listerine and jetted away.
Why does the "sanitizing light" and the water jet have to be timed? Surely the user will decide when the debris of shaving has been sufficiently removed from their razor, and will then release the button that activated the instrument? Or maybe press an OFF button. That's how my Water Pic operated: push one button to turn it on, another button to turn it off. Easy peasy, and I don't think I ever forgot to turn it off and ran the internal (rechargable) battery to depletion.
That said, you should try
@Arouse1973 IR-controlled 0.5 second timer. You will need a boost converter to obtain from two AA cells the 12 V DC it needs, but it is a good introduction to CMOS Schmitt trigger hex inverters, Schottky diodes and power MOSFETs.