Find those large staffing companies and submit your resume right away. Meanwhile keep learning and practicing. A junior HR person from that staffing company will be with you right there in the actual job interview sitting on your side of the table (but of course their customer is the company, you are the product, and they probably won't be much of a help and take a big cut). Maybe you need to practice that whole procedure a bit. Remember, if you can do FizzBuzz then everyone in the room breathes a sigh of relief since they had 3 people worse than you on this day already :-)
In the end you'll have an actual job as a real programmer doing Java at a bank and soon you'll realize the job is kinda boring and most of your colleagues are kinda average at their job and you realize why companies would advertise to find mid-senior level people :-)
But you've made it. You started kinda late in life and you wondered if you were smart enough to do it, but of course you put in the work and you got the results.
And now it's up to you if you want to cruise along or use that precious skill, the ability to sit down in the evenings and study and practice, and go on to do something you'll be proud of.
Can you do it? Yes.