I'm answering from the perspective of a mom with a daughter who takes piano lessons too, and violin (at school only in the school year, with a private teacher over the summer months).
The Queen is right -- use incentives. You mention having your daughter pay for the lessons herself, but can you see how that would only make it worse for her and she won't want to do anything at all if it costs her cash? I know how tight eight-year-old girls can be with cash.
So-- incentives. Make a weekly practice chart and work out (before you talk to her) what the incentive system will be. You need to know her currency; what does she really, really want right now that she will want in a few weeks and months? Set up something where each day she writes in the number of minutes practiced and have a daily minimum she must meet. If she makes her X minutes a week of practice, at the end of the week she gets reward Y (ice cream outing? Lego figure if that's her thing?). If she achieves her minutes for, say, three or four consecutive weeks, she gets larger reward Z (book under $5.00? One product from the smelly section of the bath store?). I suspect, from what you said about being inclined to have her pay her own way, that you might look negatively on this as bribery; well, it is. But she is so new to piano that she does not yet realize the pleasure and benefits of learning it for its own sake, so a little reward system is, in my view, a kid-friendly way to get her to practice and over time she will learn the reward of the music itself. Don't expect her to "get" that idea this early.
Get the teacher on board first. Talk to the teacher without your child around and say that your daughter is getting bored and wants to quit, and describe your practice reward system; ask the teacher how many minutes of practice daily is a minimum to go for, and get the teacher to bring up the idea of the reward chart, if your daughter likes and respects the teacher. Hearing this from the teacher, not you, might make your daughter likelier to try this.
I would tell your child too that she is going to stick with piano until at least a certain point. Again, ask the teacher. There is a kind of breakthrough point with music where a child really clicks with it (or truly does not click) and she needs to get there, and that takes time. The idea, mentioned by another poster, that piano be "seasonal" does not work for any instrument and will only frustrate your daughter much, much more as she has to relearn things!
Also, step back from the emotion of it and remember that she is not you. You had a stick-with-it personality as a kid. She may not, but that does not mean she never will stick with anything; don't let it worry you so much. She also may not have "found her thing" just yet. It's actually OK if kids do not do one or two things super-intensively at a very young age; it's kind of scary to see how kids now are extremely focused on a single activity or sport so young that they never, ever sample or try anything new or different. At least she is willing to try new things!
One other thought. Play piano music around the house and in the car, classical but also ragtime, modern, pop, whatever. Take her to see a performance. Be clear that you're not sayiing "I want you to play like this now" but you're saying, "This can be fun." And be sure the teacher is giving your daughter a variety of interesting pieces. My daughter likes to fool around with fun pieces as a diversion from the classical pieces and that helps her get in more practice time.