who is responsable for providing health care insurance for the child in a divorce?

This is one of those areas in which men object that the Child Support system is unfair to them and they are right, but it is irrelevant. At the time of your divorce, I assume there was an agreement as to which of you would provide the health care insurance (and I assume it was you). It would seem from your question that your ex-wife's situation has improved, and she would now be financially able to pay for all or part of the insurance. Unfortunately, unless she simply agrees, you have a pre-existing agreement in place.
If your "Ex" and you cannot agree on this, and you truly believe that under current conditions (changed from the time of the divorce) she is increasingly capable of providing more of the support, your sole alternative would probably be to petition the court for relief. In other words, inform the court that conditions have changed, and ask for a more equitable re-allocation of support, with your Ex paying at least part of the cost. Speak with your Divorce attorney to get the real scope of what can be done, how long it would take, and how much it would cost. It may not be worth the aggravation.