It sounds like your major complaint is with your complaining co-worker. It is noteworthy that she only complains to you, while sparing the rest of the staff from her whining. Perhaps they have been more assertive with her, and you happen to be a more willing listener.
If you are closing your complaint department, you need to be more direct with her. However, rather than simply saying that enough is enough, you can actually give her some direction as to what to do with the issues that seem to be upsetting her.
The first step is to tell her you are interested in talking with her, but not about her complaints. Then tell her that if she is serious about them, she should convert them into suggestions and present them to management.
If she actually wants to see her complaints resolved, she may give this a try. The less obvious advantage is that the act of writing out some suggestions may actually calm down her down. At the same time, she may simply be a chronic complainer who is always looking for a willing set of ears. Let her know that you are giving yours a rest.
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