If you are in Ontario, try different Service Ontario locations.
I know a family in a similar situation as your wife (passed through the border without getting reported, but do not satisfy the residency requirements), and they were all able to get coverage without having valid PR cards. Then again, this is in Ontario where provincial rules (for driver's license and health coverage) are relatively more lax, so depending on where you live, your luck may vary.
Just to echo Rob_TO, please do NOT try to engage CIC in any way. Do NOT apply for PR card, and do NOT leave the country at all. Count your blessings, and stay for 731 days without leaving for even a second before doing anything involving the IRCC.