Medicare's administrative costs are about 3%; the private health insurance industry's administrative costs are just over 20%. I don't think the problem here is with Medicare.
With all due respect, you are comparing apples to oranges:
1. Medicare patients consume more health care than non-Medicare patient's, i.e. denominator of the administrative cost/spending ratio is significantly larger. If Medicare paid the actual costs of treating their patients, the denominator would increase further.
2. The GAO and CMS admit that 10-15% of Medicare spending is fraudulent. A large component of private insurance overhead goes to track down fraud and abuse. To wit:
http://www.politico.com/news/stories/0311/50543.html
http://www.nytimes.com/2012/09/12/b...d-victimizes-patients-and-taxpayers.html?_r=0
As an aside, I once went to a talk given by George Joseph who is the majority stockholder in Mercury General, a large California auto insurer. He said that their company believed that up to 50% of their claims were fraudulent, i.e. you get into a wreck and damage your car but claim that pre-existing damage to a bumper was a result of the current accident.