J., you're in a tough spot and I feel for you. This is a really complex bundle of problems you are presenting, and to some extent, your stepdaughter is just being a normal kid for her age - losing interest in everything around her is (for many kids) part of the process of stepping back and evaluating her life now that she's getting old enough to think about it. There's an awful lot to figure out about modern life, and being a stepdaughter with a mostly-missing father could be the source of quite a bit of confusion and inner conflict for her as well.
She needs loving support, and you do, too. It's not easy (surprise!) to be the step-parent of a child going through all that. She may well be reacting to a feeling of abandonment, as are you. And jealousy, because you will, reasonably, be taking up some of her father's attention when he is around. And of course, the issue of authority looms large in this age group, as it should. Ideally, kids are learning how to internalize authority as they enter their teens, and that it tricky territory for both them and their parents.
There are a few things you can do with a child this age, and most of them start with you working on yourself.
It sounds to me like you might be enabling your husband to be an absent father and a workaholic by trying your best to take up the family slack yourself. You don't say how long you've been married, but I'll bet this situation is almost as old as your marriage, and getting worse as your stepdaughter enters her teens. If there's any way to get you, the married couple, into counseling, do it with haste. If your husband is too "busy," at least get counseling for yourself. The more you understand how to be the person you want to be, the better you can model this for your stepdaughter, and the more firmly you can ask your husband for the support you deserve, and the more likely you will have success as you undertake the hard work of helping to raise his daughter.
If you can get counseling for her, and/or family counseling for the three of you, do that, too. It helps communication to have a calm guide present as you work through family issues. That doesn't guarantee she'll respond, but it increases your chances.
If you and the foster-mother can get together and coordinate your expectations of your daughters, that could be helpful. Plus, the two of you can share some mutual support.
I could never afford counseling as a young mom, and fell back on a number of excellent books on parenting and communication. I'm sure even better ones are available now. You might check out your local library. I know it's hard to squeeze in reading on top of all your other demands, but it might be the best 20 minutes a day you can spend, helping not only with your step daughter, but your younger children as well.
And no matter how much effort you make on behalf of yourself and your step-daughter, it may not appear to help, at least in the immediate future. I spent most of a year trying to foster parent a 13-year-old girl who had some of the same behavior as your daughter. The experience seemed to be a dismal failure at the time, and I was in despair. However, years later, this young woman came to visit me with her young son, thanking me for the modeling of parenting she had received in my family. She was using that experience as a springboard for changing her life as she began the adventure of parenting. So even failure isn't necessarily final.
That was also the year I had just remarried, and my own 13-year-old daughter decided she hated her new stepdad. It was a challenging time for us all, but my husband is now a good friend and hero to my grown daughter. This would probably never have happened without constant, patient and open-hearted effort from both me and my husband (privatley, he cried often in frustration, but he was a saint toward my daughter). A lot of negative messages were coming in from my first husband, and my poor daughter was terribly confused and resentful of her new circumstances.
J., you are the only one who knows how much you can endure. And you are the only one who can decide what a good family life looks like for you. Please get help in clarifying those issues. That is possibly the best first step you can take.
Blessings, child. I will be holding your family in my prayers.