She is a car that has almost completely run out of gas. The gas is energy. Depression is having no energy.
"But I keep filling her tank up with gas!" you say. Unfortunately for you, her engine nearly kaput. Fuel leaks everywhere, the sparkplugs are bad, the throttle is loose, and despite all that gas your pouring into the tank, she's lucky if she can get a mile further down the road.
All of these things are fixable, and but you're going to have to completely rebuild the car, and since you have a budget, (in your case, the amount of time you can stand to hang around her) it's going to take awhile.
Here's how to use the car analogy to help you stay sane while dealing with someone who is, by definition, a drag. Say that she has a habit of not looking you (or anyone) in the eye. She stares at her nails, she stares at her feet, she stares out the window, but her eyes are unfocused and you know she's not actually staring at anything. This is a big habit for depressives, because by not looking at anything we don't have to care about anything outside ourselves, and therefore we can protect ourselves from any further emotional pain. It's our primary defense mechanism. So how to fix this? In car terms, the headlights are out, which is maybe why the car looks like it's been inadvertently offroading a lot. The only thing to do is to replace them. Imagining your friend as an inanimate object may seem inconsiderate, but empaths get much less worked up over inanimate objects than they do with people, so it's much easier to not take what she does personally, which is essential if you're going to be dealing with a depressed person much of the time.
So now that you have a mental defense in the form of an analogy, how exactly do you go about repairing your friend? You basically train her like a dog into certain habits. Punishment won't work, because I guarantee you nothing you can do to her is worse than what she's already doing to herself. So the two big tools in your repair kit are going to be distraction and reward. When you notice her starting to get that introspective "I'm going to beat myself up for no good reason" look, distract her. Research on the internet for things that are excellent distractions. If she's distracted, she is focusing outside herself and can't slip into full-on "I hate myself" mode. The second is reward. Whenever she does something- no matter how small- that is in the direction on forming a habit to combat her own depression, reward her. It doesn't have to be anything fancy- a smile or a sincere complement will do. With these two tools, you can slowly warp her worldview into a more positive tint.
I think that this is actually really helpful for people who are less empathic (and everyone maybe). It's still cause and effect, sort of, but just a different project than maybe you thought you were working on.