Many cancer patients resist seeking help for depression. Just the word therapist or psychologist has a negative meaning for them. Your mother may feel that her problems "really aren't that bad." I recommend using the term counselor with her, and explaining that it's very common for cancer patients to see one. In fact, that's why hospitals and clinics have on-staff social workers with special training in oncology.
If your mother strongly resists seeing a counselor, you could suggest she join a support group for cancer patients instead. Of course, she may resist this as well, as many older folks are very private and nervous about sharing personal experiences in a group. One strategy that often works is to present the group as a source of information. You might suggest that meeting other patients will give your mother the chance to hear about treatment options that you and she don't know about.
In my experience, this is the most compelling thing you can say as a caregiver to encourage your mother to get help: "It would really help me if you would go and see someone. This situation is too much for me -- I can't handle all this by myself." Your mother may not be willing to get help for herself, but if she understands that the burden is too much for you, that may matter to her more.
Sometimes, unfortunately, the only thing that works is a "tough love" approach. You may have to tell your mother, "I can't continue to help you unless you meet these conditions." Then you find a support group or make an appointment with a counselor and tell her that she must at least try it out if she wants you to continue as her caregiver. It sounds harsh, but you're actually helping her break through her denial about needing help.
Most important, though, you have to take care of yourself or you can't be there for your mother. All the studies show that nothing is more stressful than caring for someone who's sick. Even if you can't persuade your mother to get help, you can get help for yourself. Your mother's distress is probably adding to your own sense of guilt, frustration, and hopelessness, and you can seek support for those feelings.
Caring for a parent with cancer is an intense experience -- people are shocked by how exhausted and stressed they feel, and no one can truly understand except others who are in the same situation. Many hospitals and communities offer caregiver support groups, and I strongly recommend this to anyone who's caring for a parent with cancer. Joining one can make a big difference in your own mental health.