Why doesn't chemotherapy always work to treat a patient's cancer?
There can be many reasons.
The big problem with cancer treatment is that the cancer cells are your cells. They are simply faulty; the normal limits that prevent them from misdividing are removed. Cancer cells are difficult for your body to deal with because they are difficult for the immune system to detect. These are your cells, which means they have your immune markers.
Chemotherapy drugs try to target cancer cells, but since cancer cells are your cells, anything that can kill cancer cells can also kill your normal, healthy cells.