thank you for contacting the wefugees community!
The responsible authority in Northrine Westfalia is the district government in Arnsberg (Bezirksregierung Arnsberg). You fill in an application (see here: https://www.bezreg-arnsberg.nrw.de/themen/u/umverteilung_fluechtlinge/antrag_umverteilung.pdf
and attach documents like birth certificate, marriage certificate and (if you have) the "Sorgerechtserklärung". Then you send all of these to Arnsberg and wait (I guess at least 6-8 weeks, rather longer).
The district government in Arnsberg then sends you a decision by mail.
Please note that this is only general information.
In case you need further assistance, you can either contact a local refugee advisor or contact me or one of my colleagues via "mbeon". This a free app which you can download at google playstore or appstore.