Summarize:

You may not be able to solve the situation--your mother is still going to be sick no matter what you do. However, you can be there for her. You can show up when you're needed. You can offer support and encouragement. It may not be dealing with the problem as you'd liked to do, but you will be doing something, which can help alleviate your feelings of uselessness. You can pray, meditate, or just take a few deep breaths, but whatever way, take a moment to calm yourself down. Accept that you do not have control over the situation. Your mother may be sick, but you can take the time you are spending with her to develop a better relationship with her than you've had in the past. Even if it doesn't change anything, it will help others realize that they aren't the only ones who are dealing with these feelings, which is a way you can provide support. It also opens up the discussion, so others have a chance to talk about their feelings, as well. Long-term stressful situations can lead to depression, and feeling useless could be a symptom of depression, especially if it is accompanied by other symptoms.  Symptoms of depression include having trouble focusing, feeling pessimistic, not being interested in things you usually enjoy, feeling guilty, being extra tired, being consistently sad, and even experiencing physical symptoms like headaches or stomach aches.  Feeling sad on occasion doesn't mean you’re depressed. Depression is a sustained period of not caring and feeling sad. When symptoms start to overrun your life, you may be depressed. If you are depressed, you may need medication, or you might need to seek out counseling to help you work through some issues, which may help alleviate your feelings of uselessness. Remember, being depressed is not a sign of weakness. It can be caused by a traumatic event in your life, but it could also be a chemical imbalance that needs to be corrected. Also, certain medications, your genes, and other problems like illnesses can also lead to depression.
Do what you can. Take some time to stop and breathe. Focus on what's going well and make that a larger part of your life. Talk about what you're feeling with others in the situation. Check yourself for signs of depression. See a doctor if you suspect you are depressed.