If you find exercise does not rejuvenate then you it is surely depression (if its not a physical disorder) and you are better off not pushing yourself and resting yourself instead.
If you push yourself and find you feel rejuvenated then by all means that's what's needed. Often enough depression just turns into lethargy and we miss the moment to push ourselves.
As to vitamin or mineral deficiencies, only a doctors tests can really tell you if they are an issue. And it can be harmful to take more iron or other substances because you think they might be an issue. People always seem to be in a hurry to think they've got some vitamin deficiency or metabolic problem.
If you've got SAD then get one of those lamps if you think they will help.
so back to depression and fatigue, try to find ways to lift your mood. Be patient. Do it through nurturing self talk. Be gentle with yourself but try to avoid sabotaging self-indulgence i.e. chocolate and stuff like that. Take it moment by moment. find little ways to keep your spirits up. And get the rest you need.
As to the food, check your maintenance calories and just eat accordingly until you are feeling more motivated and optimistic. That way at least you might be able to avoid boredom eating or bad food habits. if you know what i mean.