Thank you for your question today, I look forward to assisting you. I have nearly 20 years of legal experience in various disciplines, including JAG.
There isn't really any method for you to simply apply to be separated. You can ask command to let you out, but they don't legally have to do so based on your request. It is at command discretion.
Now, if you just stop showing up, they technically can issue a warrant for your arrest to try and have you picked up by local authorities. That rarely happens though. The more likely result is that they will administratively separate you, in your absence, with an other than honorable discharge. I know that you stated that you don't care about a dishonorable discharge, but I think that is because you don't really know the effect that it could have on you.
You state that you are in nursing school and work at a hospital. An OTH discharge can, and often is, sufficient basis for immediate termination by an employer. Every time you seek employment, that potential employer can find out about the discharge through a routine background check.
The number 1 question that I'm asked her and in my private practice in how to get a discharge upgraded to honorable, and those people just have a general discharge. They ask because that general discharge is having a lasting effect on their ability to find work, even 15 to 20 years after receiving it.
So, while I understand that you may not care now, I think that you'll find that it is a serious hindrance to your future to have anything other than an honorable discharge in your history.
All that said, there is simply no way to force a separation with an honorable discharge in your situation.