Why do you think private healthcare isn't working? Free market is typically inefficient if 1) there's not enough competition or 2) customers don't make optimal market decisions. Which one is it in private healthcare?
First, it's difficult (though not impossible) to price-shop for medical care. Then sometimes your medical care is time-sensitive in a way which few other purchases are: if I slip and cut myself badly, I'm not likely to price-shop for my ambulance, then carefully compare emergency room price schedules, then gather bids from various doctors who'll be operating on me...
Also, "health insurance" distorts purchasing decisions by hiding costs -- if your insurance will pay any doctor you go to, and its only cost to you is some abstract impact on the rate your employer pays for their group insurance plan, you're not terribly motivated to be price-sensitive.
You could overcome all these things by wildly changing the way our system currently works. But if you're doing that anyway, I'd much prefer a government single-payer system.
I am Canadian, but living in the States. I have experienced both systems, and I can say unequivocally that I vastly prefer the Canadian socialized system. The quality of care that my family has received has been about the same with very little out of pocket in either case, but in Canada, I simply don't have to think about healthcare, ever. It is simply there for me [EDIT: even if I am unemployed].
Source: http://kff.org/health-reform/state-indicator/total-number-of...