I read there was a fairly good case for such a doctor by the name of *Patrick S. O'Reilly, *who had strong sadistic tendencies, and a missing pectoral, which fitted the lacerations on her body, and relation to police (he was formerly married to a daughter of a police captain), which could have greatly assisted in a cover-up. Also were there any murders with similar modus operandi and patterns in adjacent years/locations?
The short answer to your question is "Nobody knows who murdered Elizabeth Short".
A lot of theories have been advanced over the years, and several suspects have been named. Patrick S. O'Reilly is certainly one of the names that has been advanced as a suspect, but it seems that it was actually George Hodel who topped the LAPD’s list of suspects at the time of the crime. Hodel's son became a detective in the LAPD and has published much of the 'evidence' that he believes confirms his father's guilt.
As for other similar or related murders, a few have been suggested, but none with entirely convincing evidence to support the link.