I know exactly how you feel, we're dealing with almost the exact same problem. We homeschool our kids, but had to start one in a small private Christian school this year also, for the exact same reasons! He's also in 5th grade and just turned 11 (his twin brother still homeschools).
Here's how I/we are looking at it, and what we are doing. First of all, we have him at a Christian school because those are the values and "worldview" we want him to have. We put him in school because he needed more consequences for not getting work done, and for it to be on HIS shoulders, not mine! If you nag your daughter or follow up with her constantly to make sure the work is done, and to your satisfaction, that burden is on your shoulders, not hers. She will not do the work, or pay attention until you get to her breaking point.
What you want to do (and we're doing) is get her to carry her own burden. Here's how we're doing it...
1. We clearly laid out expectations for grades. His teacher already explained how grades are calculated.
2. We explained that we will not nag or ask a bunch of questions about his work, just expect him to meet our expectations.
3. His homework and studying are HIS responsibility, if he truly doesn't understand something, he may ask us, but if we have already explained it, or believe he's just wanting a crutch, he's on his own.
4. We told him if grades were not up to standards, we'd make sure he brought them up, and he wouldn't like it. (He'll be in his room except for school, chores, meals, church; no more basketball, playing outside, etc; and he'd have a new, earlier bedtime) We didn't tell him the exact solution, but promised he wouldn't like them. (you can tell her the consequence if you choose, but it must be drastic and facilitate more study time).
If he does not keep his grades up, he'll have one month (or mid interim report) to bring them up. If he's raised them, he'll get to resume "fun" activities. If not, he'll have until the report card to fix it, etc. He will remain in his room until he does it. Now, we already know what is capable of, and expectations are based on his abilities, not his current effort! We're not asking him to do something he's unable to do, just choosing not to! We want him to make better choices.
Again, if I nag, cajole, or constantly "motivate" him to keep grades up, he'll NEVER learn how to do it on his own! That will translate into problems in a job, paying bills, and basically maturing into an adult. Some people may think I'm mean or too strict, but I'm thinking longterm. Hope it helps.