Truthfully? There are plenty of savings to be made, but I'd highly doubt there are any cheap answers. There are several different kinds of socialized medicine, and where broadly implemented, are all expensive. In addition, there are a few additional costs that are usually overlooked:
Firstly, if you have money, American health care is really excellent. There will be declines in the quality of services provided, in at least some areas, for some people, with the advent of a more egalitarian system. This will also hit people lower down on the quality scale than you might expect, at least in part, by really rich people not being soaked as heavily.
Secondly, medical research is an extremely difficult field of human endeavour. There is a very strong view today that medical research is cheap and only held back by legislative difficulties or fat profit-taking capitalists. That view, by and large, is untrue. Finding drugs and new treatments for humans is really, really difficult and really expensive. The easy ones have been found. The difficult ones have been found. The targets people are going at today are extremely, absurdly difficult problems; New technology helps, but in general, progress in medical research gets harder and more expensive every year.
And there's no getting around it. Someone has to pay the bill, and the result is that everyone pays. Americans pay a lot due to there being a lot of them, and generally being wealthy. Countries with rich socialized medicine also pay lots, although generally less than Americans. Poor countries pay too, albeit less.
The biggest chunk of the expense is also the one that can never by bypassed. Clinical trials. They're crazily expensive and absolutely necessary. At the most basic, new discoveries are made by exploring the unknown. That is, we are going to find new things by doing things that haven't been done before. As a result, it is absolutely necessary to test said discoveries in order to learn if they work.
Cutting off the rest of the world doesn't make the problem better for Americans. It means they'll have to foot 100% of the bill, instead of the current existing large percentage. It is also one of the places where socialized medicine should help, by being able to cut out some middlemen in the process and more efficiently providing money for new drugs.
But it won't be cheap. Cheap would give you health care that doesn't make people better.
..Also, ten trillion dollars over the next decade seems pretty reasonable. There's what, 450 million americans? One trillion a year is a bit over $2000 per American, which isn't an unreasonable estimate.