Oh my gosh, your story is so similar to mine.  I haven't met you but I can tell you are not a useless mother.  I too am divorced, have 2 pre-teen girls who constantly argue with me.  All I think is "If I talked that way to my parents they would have spanked my butt red", I didn't want to raise my girls that way.  We had a horrible divorce, 2 custody battles and after 10 years still doesn't pay me child support.  I tried to compensate with them too.  I also got very, very depressed.  Never felt like that before.  They know how to push my buttons (now they do, not when they were younger).

I feel for you, I wish I had some awesome advise to tell you, but remember your not alone, I'll be sending good thoughts your way.  Just hang in there, stand your ground, calmly but firmly tell them this isn't ok and even if it hurts and you don't have the energy tough love it.  After all we are the moms right?  We brought them into this world. 

My girls just started being rude (especially the oldest) she's a month away from 13.  Every morning, and when I pick her up from school, she is sarchastic, rude, yells at me!  She was my angel until about 4 months ago, now she's demon child. lol  Now my youngest 11 is defying me and she always, always minded me.  Myself, we're going to get counseling, maybe that might help you, but go as a family.  I don't know...I'm trying to help.