Am I as a non-custodial parent, required to provide insurance for the custodial parent?

First of all, you can not legally have some one covered under your plan that you have no legal relationship with, and you have no legal relationship with your daughter's mother. Further, if you let her use your wife's insurance card then you would be committing insurance fraud - not something I think that you want to get involved with here is it?  Reimbursing you for the bills has no bearing on any issue here so it need not even be addressed.  What she needs to do is inquire about obtaining COBRA benefits based on her prior insurance coverage before they terminate coverage at her place of employment or buying insurance elsewhere.  Your daughter, however, can be placed on your policy if she is not already there on.  Good luck.