I am personally not a fan of the humor method...I would much rather have someone be direct with me. I don't equate communication with confrontation.

But, if there is a lot of drama there, and people tend to have the low self images that often go with that, what about a little self-deprecating remark to help diffuse her defensiveness?

"You know, I realize I might be a little anal to be this bothered by your gum...but for whatever reason, I am, and I would be really grateful if you would chew just a little more quietly, as a favor to me?"

I happen to have extremely sensitive hearing, and I am anal enough that I let stuff get to me. So, I have both an iPod and a "natural sounds" machine in my office. If I just reset my mental focus, and then listen to something I like, normally the sounds of others recede (out of my zone-of-irritation, if not actually out of my earshot).

Good luck!!