I think finding a PT that would work w/ your DDs' school hours & in which you would net at least $600 is going to be really hard to find.
Like others have said, a PT job that works w/ your kids' schedules & is flexible enough so that you can take off when your kids are sick or have activities or have school holidays is almost impossible to find.
In our case, sometimes I think I would like to find a part-time job just for a little bit more spending money. However, DH works 12 hour split shifts. Where he works is an hour away from where we live, so, on his work days, he is gone 14 hours/day. Additionally w/ his rotating schedule, he has 2 weekends off & 2 weekends where he works - his schedule changes from week to week (he's not off the same days or same # of days/week), so there's no way I could find a job that would let me work on his off days - since my available days would change week to week.
Someone has to be here for the kids & their activities. We homeschool right now, but, even if we put them back in school, someone would still need to be available to take them to school, pick them up, get them to their activities, be home on off days & sick days & during the summer, etc. Now that DD is 13, leaving them home is a little easier, but still she can't drive yet!
Additionally, if I were to go back to work, we'd need childcare for our youngest - what I could make at a PT job, I would be turning right around & paying a childcare provider.
Plus, in our own situation, w/ my DH's crazy schedule, one of the reasons we decided I would stay home is to provide the kids w/ a stable & reliable routine - essentially, it's my "job" to keep things moving & "normal" here at home, if that makes sense.
I realize that your nephew is possibly a handful, but do think seriously about the amount of flexibility you have now that you would be giving up if you began outside work. Since he's your nephew, I would think having him would almost just be like having a 3rd child (somewhat different than keeping someone not related to you), & there would be ease in taking him w/ to your daughters' activities, etc. W/ my nephews & nieces, I would feel more comfortable integrating them into my family's schedule & "treating them as one of my own" than I would w/ a child that wasn't related to me.
Do you really want to give up the freedom & flexibility you have now? Can you?
Once you factor in all the different variables (taxes, gas money, work clothes, sitters for your daughters on sick days, etc.), can you clear more than $600/month at an outside PT job? If you can, I would consider it. However, I think I'd also think strongly about asking my brother for a raise first.