There is no tax procedure to fix this. There are some things you could attempt but they would be red flags for an audit, and I would not mess around with this problem unless you hire your own tax professional who will represent you if audited.
I think you need to go after the employer. If you enrolled your spouse during the proper open enrollment period (or proper life event period) and with the correct documentation, I would think you have a legal claim against them if they don't fix the problem. And they certainly can fix it, both on their tax filings and your W-2. It's a bit complicated and if it is a small employer they might not want to bother, or if they are using a payroll company they might not want to argue with the payroll company. And fixing this will save them money as well, so I don't understand why they won't fix it, as long as you submitted the proper documentation when you enrolled. But they certainly can fix it if they want to. If the company won't help, I would contact your state labor regulator.
Of course, if you anger your employer you might be out of a job. That's a tough calculation to have to go through. Fight, maybe get a lawyer, or try to backdoor it on your tax return and hire an accountant to help.
