My compost pile is completely surrounded by huge nettles making it so difficult to turn. I'm hoping to get some nice soil from it for mulching. Are you keeping the nettles for butterflies? If not, add them to the heap, they're very nutritious. Cut them off and add the green bits. Ideally you should dig out the roots - yellow, non-stinging and very tough, so you can pull hard and get about 6ft of root out - very satisfying. But if that seems daunting, then repeated cutting of the tops will slowly weaken and kill them.
Nettles are also one of the "can be eaten like spinach" plants that are actually worth eating. Pick the new tops in spring. They'll lose their sting when cooked, or you can blitz them with milk in a smoothie aker and add seasonings to make soup.
Which plants are the ones that need mulching most to survive /thrive? Anything that's a greedy feeder. Rhubarb. Nettles 
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The other advantage of mulching is that the soil gets more friable and it's much. much easier to pull weeds out.
Try putting some big old flower pots or other containers around the garden. Then if you can persuade yourself into a walk round the garden every day (which will itself help to lift spirits), you can pull a couple of weeds or dead head some plants as you go around, and dump them in the nearest container, to be added to the compost heap when you can be bothered. It's amazing what a difference you can make in a garden in 15 minutes a day, done regularly, and every achievement helps in the battle against depression. But don't beat yourself up if you miss a day!