Your priorities are messed up if you’re complaining that a student’s questions “zap you of your enthusiasm.” Teaching is (or at least, should be) about addressing the student’s needs, not the teacher’s. Enthusiasm should be generated through interaction with your students, not from your satisfaction with your lesson.
My suggestion: try legitimately addressing the points she raises. Don’t write them off as mockery, even if it IS clear that she’s only questioning out of belligerence. Don’t try to divert her with unrelated incentives. Don’t be defensive or evasive if you can’t give her an answer–say, “That’s a good question, I need some time to think about it,” and try to prepare something that SHE will find meaningful. If she’s putting a negative spin on everything you say–try to look past the negativity, pinpoint the issue she’s bringing up and open it for discussion. Whatever you do, be straightforward and level-headed, and don’t let your pride or personal outrage get in the way. If you do that, I think you’ll find that she will respect you more and might even be willing to listen to what you have to say.
Also, in general, try to have an bit of leeway for an open forum during the lesson. Don’t try too hard to keep to your script. Mentally prepare yourself for spontaneous discussion, and encourage it. Don’t make this about “fixing” this student–it’s really an opportunity for you to develop better teacher skills and genuinely engage your students.