Measuring customer satisfaction

I work for a alcohol and drug rehabilitation centre. It is difficult to measure customer satisfaction. We do have a customer satisfaction questionnaire that we ask our clients to complete when they leave however in a number of cases they don't want to complete the questionnaire. Or they leave before they are done with rehabilitation (for whatever reason) and they might give a complete irrational score. With regards to customer satisfaction, do we for example have to include this as a target...to improve customer satisfaction? I am afraid that we might set ourselves up for failure if we do this. Or would it be acceptable to show evidence that we obtain feedback where and when possible, and then, if there were any suggestions for improvement, if we log these as suggestions for improvement in our electronic QMS program.