I seriously feel like I should report that lady at work to HR

When you're my age, you'll know this. HR is not there to protect you from the organisation. It's there to protect the organisation from you. If occasionally your interest coincides with HR, they can be useful. If the lady is senior to you, and particularly if your manager, HR will show her how to legally dismiss you. They may even engage in outright illegalities, such as making you redundant or contriving conduct charges.