Think about your skills. What skills do you have that will help you get your internship and your job? You seem determined to get this and I can only assume that you're not a slacker, so there are two good things about you. Maybe you should work on getting some new skills that will help you be successful. Express interest in the job. I mean, I've only sent in a few applications, but maybe you could tell them that this is your dream and how long you've worked on getting here and how long you've dreamed about this.
Don't take rejection so hard, okay? Just send them to as many as you can and, if one falls out, then know that you have several more backups

. Keep sending them out and, eventually, you'll get the internship. Also, rejections aren't always because of you. Maybe they just ran out of space, maybe something unexpected came up, there are a lot of reasons for rejections, so don't be so hard on yourself, alright? You've got what it takes. Show them.