There are two problems that still remain.
1) How are you going to enforce the individual mandate?
2) How are you going to stop (or mitigate) "churning" in the system?
These are inter-related problems. Health care insurance is still dependent on your ability to hold a job. Labor turnover, especially when the economy is doing well, is high. People come and go from positions. Should we punish people that only have insurance for half the year? Or punish companies because they don't want to hire (or keep) someone because of they do not want the health care liability? Health care companies can't discriminate against you, but employers who are responsible for a share of your premiums still will (legally or not).
Here is the legislative change that I propose, call it "How to fix ObamaCare":
All health insurance plans go for a single year with the same start and end date. Call it April 15- Tax Day. If you get fired, you still have your health care until April 15th. Everyone's new plans starts on the 16th.
Beginning in January, the exchanges open and everyone can buy health insurance for the coming year. Employers can buy it for their employees, individuals can buy it for themselves, people can sign up for Medicaid, Medicare recipients can choose which Medicare plan they want to be on, etc. All of this can be done online. For those who are not adept at technology, then at the Post Office. I am a big proponent of Post Offices being hubs of all your Federal Government needs.
Your physical insurance card will be mailed to you as soon as you sign up. An email copy can be sent to you in the mean time. When you file your taxes, which are due on April 15th, then you enter your health insurer and group number to prove that you have insurance. If you don't have insurance, then you pay the penalty that is specified in the mandate. If you don't file your taxes, then you lose your refund - which pretty much was the only penalty that the government could assess anyways. Most health care subsidies and reimbursements happen through the tax code, anyways.
Both problems are taken care of at the same time. The government can ensure that you have insurance; enforce the penalty for those who don't. And people know exactly when their insurance lapses. Churning is mitigated substantially.
Now their are going to be some distortions of the labor market between January 1st and April 15th, because employers might want to hold off hiring or speed up firing so they aren't on the hook for health care for a full year. Some regulations could probably correct these distortions. Here is an example, but it could be a whole grab bag of policies: Make an exception that the employer does not have to cover health for the first year of their employment if they are hired during these months. The individual would be eligible for the same subsidy on the exchange as if they were employed or unemployed during this period. On the flip side, the employer still has to cover health care (with the help of the COBRA program) if they were to fire the employee during these months. This would smoothen hiring and firing for health care related reasons.