As an employer, you are required to collect that information when you hire the worker, even when it is a household worker. However, generally casual labor paid for in cash of small amounts can be handled on a contract basis, and individuals do NOT have to issue 1099s. The problem is if you have more than one household worker, you generally are going to be viewed as an employer regardless.
As you say, you cannot report these taxes without having the employee's information. You need to delete all traces of any worker on Schedule H for which you are not reporting/paying taxes. This can be a little trickier online, but one approach is to delete the whole Schedule H and then start over (if you have other workers to report.)