It'd rather seem that Jefferson really didn't have the answer either. His 'solution' probably wasn't a solution, and would probably drive said paper out of business.
Yes, and I don't believe there is a definite answer. I like his description of the problem, and the fact that it works very well to describe the problem today is noteworthy as well: it's not a new problem, but each generation has to deal with it, and maybe the magnitude changes from time to time.
The answer is either private organizations have to act with integrity to promote and coordinate 'proper news' - or there has to be social intervention. The former kind of holds true for network news wherein there is institutional control over distribution. But not so much in print, and definitely not on the net. In Canada, there's the CBC and they are considering subsidies for other news outlets.
Relying on private organizations to act with integrity isn't going to be easy, I think. For public options, I'm skeptical as well, they are very mixed in my perception, some are doing good work, others are little more than very expensive mouth pieces for the government. Subsidies for private companies might be a way, but that will undoubtedly get gamed and you end up needing a large bureaucracy to counteract that.