How should I handle a grumpy Soldier?

SFC (Join to see) IN a non-accusatory manner, neutral location away from other staff ears... Sure, I would. "Hey, noticed the office and staff really frustrating you these past few weeks I've been here... would like to know what the issue is . Cant help make it better if I don't know what is wrong"

Find out what is causing her disdain. I mean, stellar performer but expressing issues. There is an underlying issue. As a Leader, it would behoove (and I really do hate that word) you to try and find the root cause.

On the civilian side, I had an employee that was the hardest working guy in my department, never missed, could be counted on to get things done correctly, but , using the infamous B word, he just bitched constantly about everything. After I had been there about a or so month, he asked to speak to me in private, where he proceeded to gripe about the situation in the world, the company, the local city and life in general. Anything to do with work I told him I would look into and into his solutions, and if that was all and if it was, go on back to work. I soon found that this was going to be a regular meeting, usually a couple of months went by before he had a enough gripes built up for another meeting, then he went back to work just like before. His work related comments were generally on target, or at least relevant, and he often had a well thought out solution. The rest of the general griping wasn't really focused that well, and I think he just need someone to vent to occasionally.
After I left that position, we became friends and hunting buddies until his death. He still complained about everything, so much so that some of his family had nicknamed him Bitchard (for Richard, of course).But my Master Sergeant Father always told me it was a soldier, sailor or airman's right to gripe about things. The point of that story is why don't you just take her in the office and ask her if she needs to get things out that are bothering her. If it's something that you can address, try to do so, if it's general I am not happy with the World, offer so sympathy and a ear to listen to her, then send her back to her duty. If it is effecting section efficiency, just ask her to bring it to you directly instead of venting in the section.
I had a Platoon Sergeant that had a similar attitude, but otherwise was a solid NCO. I listened to his gripe session everyone once in awhile, which isn't that big of a price to pay for a good performance. Like my daddy stated, soldiers are allowed to complain.