The first step is to immediately try to learn from this experience: why was there a surprise right at the target date and not before? The point of a mitigation plan is to define a path for addressing the issue and monitoring progress all along the way. So for next time, consider whether there should have been
interim check points in the mitigation plan to assess progress, rather than assuming the mitigation plan would automatically work, or simply assuming that the functional group handling the risk would let you know when it was done.
Given where the project is now—with an outstanding risk that hasn't been resolved by the target date—what's next? Review the current status of this risk. Has it been partially resolved such that the risk has lessened, but we're not totally out of the woods? Is it still at ground zero, with no good solution in sight? Based on that status, assess the potential impact it still presents to the project. Does this item still represent a serious threat to the schedule, costs, or scope?
Then it's time to decide whether to move to a backup plan. During risk assessment and mitigation planning earlier in the project, the team may have identified backup plans for specific risks. "If this item isn't solved by this date (the mitigation date), then we'll move to Plan B," (for example, to punt back to using a less risky technical solution). In this case, the mitigation date serves as a "trigger date" for invoking the contingency plan. That date should have been set such that the team had time to move to the backup plan and still have time to meet all or most of the project goals.
Even if you didn't identify such backup plans early, consider them now. Can you adjust the plan to remove the risk? What would be the impact to other project goals of doing so? For example, reducing scope or adjusting the schedule slightly to provide a bit more time, if the issue is seen to be nearing resolution. Tools you can use to help you through this process are the Planning and Scheduling: Make Trade-offs and Optimize template for assessing the impact of alternative approaches, and the Flexibility Matrix to help guide tradeoff discussions on scope, resources, and schedule.
In summary, the task is to assess where you really are and decide whether and how to give this risk a little more time and effort to resolve it, or to move to a backup plan that will eliminate the risk and still achieve the most important project goals.