The reason lifting weights helps treat depression is because you don’t just work out your muscles. You also exercise the ability to dispense motivation. Build up your power to motivate yourself, you’ll find yourself doing little things that need to be done. It’s not perfect, and it can be a really slow start, but it really does help.
It’s a lot easier to lift a dumbbell a few times than it is to clean your room, but if you push yourself to lift it you will find it easier to push yourself to organize your stuff or do the dishes.