You have some options, but there are a few things to be aware of upfront:
First, confirm that the issue was actually on your employer's side (if possible). If it's an internal list of emails that HR compiled and a staff member mistyped your email, that's the HR department's fault. If, on the other hand, you filled out a form for your company and mistyped your own email address, which was then blindly passed on to this process, then blaming your employer will be tricky at best.
Second, understand that given the structure of your employer's contract with this insurer (and the nature of the health insurance business in the U.S.) it may not be possible to get coverage through this insurer or access your employer's plan for the upcoming coverage year. A given insurer might flex the rules (I've seen it many times in many situations), but they generally don't have to-- there is no "oops" provision in insurance contracts.
Third, your employer may well take the position that you had some responsibility to follow up on apparent issues in the process. If they announced a coming email with new insurance enrollment information, and you didn't get any such email for a few weeks, they may say that you should have inquired further. Whether or not it's actually reasonable in this case, it's not unlikely that your employer will say you had some agency and responsibility here (even if the mistake was not yours).
Fourth, your employer may have budgeted $X per employee with a given insurance option, reflecting premium rates offered by the new insurer for all of your employer's staff. Even if they were to give you this money in cash, it may not match up well with rates offered on plans that are actually available to you (such as a policy purchased through an ACA exchange).
So, with those out of the way, the situation is this: based on what your plan administrator said, this insurance policy may be out of reach for you until next year. This is a bizarre enrollment period (those are typically September - November or so), but if that's what the contract states and the insurer is not willing to cut you any slack then you will have to look elsewhere for coverage. I personally would press the plan administrator a bit harder on whether or not the insurer is really unwilling to offer leeway, but it could be the case.
Employers in the U.S. don't have to provide health coverage directly, though they typically have to offer it or extra pay as a voucher for you to put towards an individual health plan. It's worth pushing that angle, since your employer's apparent mistake has deprived you of either option. And, since they aren't paying any portion of a premium for your health insurance, the money they had budgeted to do so is just sitting in their accounts. It may be a relatively easy shift for them to give you the extra money as long as you can demonstrate that it's going to health insurance premiums.
Do some research on health plans available to you as an individual consumer. The ACA marketplaces are a good place to start, but you can also talk to insurance brokers in your area. Coming to your employer with a solution and hard numbers may be more persuasive to them:
Because of a clerical problem from HR, I've been shut out of the company health plan. I found this insurance plan, which conforms to ACA rules about obtaining coverage not offered by an employer, which would cost $X per month. Since the money originally budgeted for my premiums through the company plan isn't being used for that purpose, I'd like to apply it to this alternative plan until things can be sorted out during next year's open enrollment period.
- It is entirely possible that you will not be able to join the company
health plan. This is ultimately up to the insurer and may not be
something that your employer can affect at all.
- It's worth pushing your plan administrator on whether or not the
insurer truly won't accept you into the plan, and if so you may be
able to get some additional money from your employer to put towards
premiums for a different health insurance plan.
- Finding an individual insurance plan will likely involve higher
premium rates for similar coverage.
- Your employer will likely argue that you had some responsibility to
be proactive in this process, and by not doing so have implicitly
declined insurance coverage or at least are partially to blame for