How did Holmes reveal the identity of the perpetrator? Chapter 12

In Chapter 12, Holmes tells Watson that there is record of a relationship between Stapleton and Laura, leading Holmes to believe Stapleton the culprit. Holmes has uncovered that Miss Stapleton is actually Stapleton's wife, not his sister. This is why Stapleton so vehemently opposed any union between her and Sir Henry. Through research, Holmes has learned that Stapleton was indeed a schoolmaster who lost his job, but that he had operated under a different name and then disappeared without a trace. Holmes has deduced that not only was Stapleton the bearded man in London, but that his wife must have been the person who tried to warn them. Clearly, Stapleton believes he can gain some benefit from lying about his wife's identity.