I am not a doctor, of course, but I strongly urge you to bring this problem to the attention of your doctor. Don't be afraid or embarrassed to do this, and don't feel that it somehow shows a lack of faith on your part—for that isn't true.
From what you say, your wife sounds like she is seriously depressed, and your doctor should be able to evaluate her situation and suggest what steps you need to take. Depression can be caused by a number of different things, I understand, including chemical imbalances in the brain that can be treated medically. Remember, God knows all about your wife's situation, and He wants to help her get the best treatment possible.
Does this mean prayer isn't important, or that it doesn't work? No, not at all. God certainly uses doctors to bring healing to us; their skills are a gift from Him. But prayer is also important—even crucial. The Bible says, "Pray for each other so that you may be healed. The prayer of a righteous man is powerful and effective" (James 5:16).
In addition, ask God to help you be the kind of husband your wife needs at this time—thoughtful, loving, kind, patient and understanding. She needs to know you and your children still need her and love her, even if they aren't living at home. Most of all assure her that God still loves her, for His love never fails.