In "A Case of Mistaken Identity" how does Holmes discover that Mr. Windibank is Hosmer Angel?
In "A Case of Mistaken Identity," the evidence of Hosmer Angel's identity started building with the revelation of the motive for the mystery: Miss Mary Sutherland has a small fortune that she lets her mother and young stepfather use until she marries and needs her money for her own home and family. The evidence mounts with the peculiar coincidence of Hosmer Angel's appearance only during the stepfather's, Mr. Windibank's, absence. The peculiar description of Mr. Angel's traits adds more grains to the mounting evidence, his "hesitating, whispering fashion of speech," his tinted glasses, his preference for evening strolls.
The apex of the mounting evidence comes with his typewritten letters, with typed signatures, and with his insistence that her letters be sent to general delivery and that she swear to be true and wait for him to claim her pledge of love to him should anything unexpected happen to him. Once Holmes had tricked Windibank into sending Holmes himself a typewritten note, the culprit was caught with the turn of a key based on the comparison to more than fourteen distinctive characteristics visible from the strikes of the typewriter letter keys.
As Holmes says, no two typewriters produce the same quality of type. The e and the r in the love letters from Mr. Angel to Miss Sutherland and in the note from Mr. Windibank to Sherlock Holmes had the same fourteen plus characteristics and therefor were typed on the same machine by the same man, Miss Sutherland's stepfather, Mr. Windibank who, along with Mary's mother, was playing a very unkind trick on Mary for reasons of selfish gain from her monthly income.