As there is no way an educator could know your individual feelings or circumstances, or what sort of information was being disclosed, the most helpful advice I can provide has to do with how you should approach and structure your speech.
The introduction should give the audience a sense of the purpose of the speech. In other words, you should think about why your audience would be interested in hearing about this. You should think also about whether there is a moral lesson you are attempting to convey, and whether the point of your speech is that it is bad to betray confidences or that it is imprudent to over-share personal information at work. The main purpose of the introduction is to capture the attention and sympathy of your audience.
The second section of your speech is what in classical rhetoric is called the "narratio," or "narration." This should consist of a simple and factual recital of the major events that occurred, including the type of information shared and the degree to which you made explicit your desire that the information remain confidential. You should also mention why you wanted this information to stay confidential.
Next you should describe the moment when you first discovered the information had been shared at work. A good way of organizing this part of the speech would be by the individuals with whom the information was shared. In other words, you might state, "When I arrived at work the next morning, the [security guard or receptionist?] mentioned..." Then you could describe how that made you feel and why. You could talk about how this new, shared information affected your relationships. You might continue in a similar way to discuss interactions with several other people at work including managers and friends. You might conclude this section with a discussion of your first meeting with the person who betrayed your confidence after you found out about the betrayal.
Your conclusion might reflect back on what you would have done differently.