being a step mom

So I have to rant. And get some opinions. 
*long* 
I'm extremely worried about my fiancés daughter. She lives with her mom and we get her every other weekend and most days during the week. Her mother to me is a very unfit mother... she barely takes her daughter to school. I'm talking like maybe 2 times a week and sometimes she will miss weeks. We have been notified by the school because all her school stuff gets mailed to his house (I live there) so we keep all those papers. She is 6 years old. Can't write her name right all the time. Can't count past 30. And can't write numbers past 12. And half of those number are backwards. She can't read she can't barely spell and doesn't know all her abcs by writing them down. (She's in the first grade). Her mom does not help her with work or do anytype of homework with her. Recently me and her tried to be civil with each other. She was fine with me until we got engaged, after she found that out she was pissed and stopped talking to me. Extremely childish. Last Monday I went and picked up his daughter from school I tried to do. Work with her because she told me she wanted to be in the 100 club in school. So I tried to practice with her. And it's just hard because she doesn't understand. So I told her mom that I was gonna start doing flash cards with her and practicing sounding out letters and work on her letters. Basically practice EVRYTHING with her. Ever since that we have not seen her nor have her after school. My heart breaks for this child. She is a lazy person. Her house is a pig sty. She can't keep a job. She can't keep a home. I have been begging my fiancé to go to court and fight for custody. And full temporary custody until she can get her life together. This child NEEDS to go to school and Needs a stable house. We also just had a baby. So I couldn't imagine not teaching my child everything she needs to know. Instead her mother is worried about buying her expensive things and Victoria secret stuff ! I'm just so tired and fed up. I finally broke down and cried to my fiancé and he is now going to court and got the papers to file. Am I in the wrong here ? I just need outside opinions.