Thanks for clarifying the matter!
As far as I can see you have to options to be able to stay with your family:
1) You two can try to get married. That would grant you the right to stay in Germany and also to work here. But since you are a rejected asylum seeker that's risky because you need to hand in your passport at Standesamt (registry office). If Ausländerbehörde gets access to your passport, it's easier for them to deport you.
2) If No. 1 is not an option you can try to get acknowledgement of paternity. In a nutshell, you have to declare that you want to take care of your child together with its mother and then you will get a residence permit in return (it's called "Aufenthaltserlaubnis zur Personensorge").
Please check out the following link for further information. Also keep in mind the additional information provided in the comments: