If you have that much hair in this area and new hair comes in every day, it raises a red flag for an underlying hormonal disorder. These disorders have far reaching complications, in addition to making hair grow in all the wrong places. If I may suggest that you see your endocrinologist to conduct appropriate hormonal testing. If your hormones are not in order, you may experience more new hair growth in this, as well, as other areas. Electrolysis will permanently remove the hair that’s already there, but abnormal hormonal levels may stimulate dormant hair follicles to grow new hair. Luckily, this condition (if you have it) is treatable. Still, if it is left untreated, it tends to progress itlesf. On the other hand, if your hormones are in good order, you’ll, at least, feel relieved.
You should definitely consider electrolysis in either case. The area you describe is not very big, and I have seen drastic improvement after just a few sessions. By “drastic improvement” I mean that with a face that took 1.5 - 2 hrs to clear initially, after a few sessions, the client comes in every two months for a 10 min clean up. (And the 10 min include the eye brows, that weren’t initially treated). This client used to shave regularly, but now, as James has put it above, she only washes her face in the morning. She is not totally finished yet, but only she and I know this. In everyone else’s eyes she has a beautiful clear and smooth skin. What a nice alternative to facing yourself in the mirror every morning, while reaching out for the razor, and worrying if someone has noticed your hair throughout the day!
You face may take even less time, the above is just a recent example. An important point to stress, however, is that you must find a competent electrologist to do the job.
The best of luck!