First of all, I am really sorry to hear about your difficult situation and your girlfriend's accident.
I am afraid, however, that the immigration office will not issue a residence permit for you at this stage to take care of your girlfriend. Travelling within Germany (and without papers) might be possible, but is definitely also a risk. Would you need to travel far (and with public transport)? I really don't know how we can help you here. I would highly recommend you to get some individual (emotional) support in a counselling office. You don't need to worry about the contact - there are many organisations which offer free of charge and confidential services.
Regarding you girlfriend: I hope that there are also family and/or friends who can visit her in hospital now in case you are not able to go!?
If you want to share your approximate location, I would be happy to make some research on counselling offices in your area and contact details
All the best for you and your girlfriend,