I think I can be a software engineer

I have no doubt in my mind that you can. Keep at it.

> ... not everything I have ever done has been a success

> ... algorithms are challenging

> ... new javaScript framework every week

> ... thousands of other men and women trying to get jobs

This may sound rude, but -- no one cares about any of this. All you need is more experience IMO and creating websites or writing software to scratch your own personal itch will be your best investment.

> ... willing to keep trying ... until I die

Okay, that's probably good. Don't give up.

> build using modern web technology ... Using tools like Grunt, Git, GitHub, JIRA are second-nature

You mentioned github but don't include a link to your account or any repos on your resume. If you haven't published anything, take time to demonstrate some competence and create a public repo. It will serve as a portfolio that employers can review.

> Bachelor of Science in Business Administration

Wow -- BusAd? A university degree is so expensive these days. Why spend so much money getting a degree that intersects so little with your interests?

Well, regardless, it doesn't matter. For my degree, I learned about automata and all kinds of other stuff that I so rarely apply to work. Just get experience working on software and the career will follow.