There has never been a shred of evidence that the Chicago(or any other Mob Family) was involved in the JFK killing.
As to the House Select Committee, their "investigation" was a joke. Check out the conclusions that they reached:

On the Kennedy assassination, the HSCA concluded in its 1979 report that:

Lee Harvey Oswald fired three shots at Kennedy. The second and third shots Oswald fired struck the President. The third shot he fired killed the President.
Scientific acoustical evidence establishes a high probability that at least two gunmen fired at the President. Other scientific evidence does not preclude the possibility of two gunmen firing at the President. Scientific evidence negates some specific conspiracy allegations.
The committee believes, on the basis of the evidence available to it, that Kennedy was probably assassinated as a result of a conspiracy. The committee was unable to identify the other gunmen or the extent of the conspiracy.
The committee believes, on the basis of the evidence available to it, that the Soviet Government was not involved in the assassination of Kennedy.
The committee believes, on the basis of the evidence available to it, that the Cuban Government was not involved in the assassination of Kennedy.
The committee believes, on the basis of the evidence available to it, that anti-Castro Cuban groups, as groups, were not involved in the assassination of Kennedy, but that the available evidence does not preclude the possibility that individual members may have been involved.
The committee believes, on the basis of the evidence available to it, that the national syndicate of organized crime, as a group, was not involved in the assassination of Kennedy, but that the available evidence does not preclude the possibility that individual members may have been involved.
The Secret Service, Federal Bureau of Investigation, and Central Intelligence Agency were not involved in the assassination of Kennedy.
Agencies and departments of the U.S. Government performed with varying degrees of competency in the fulfillment of their duties. President Kennedy did not receive adequate protection. A thorough and reliable investigation into the responsibility of Lee Harvey Oswald for the assassination was conducted. The investigation into the possibility of conspiracy in the assassination was inadequate. The conclusions of the investigations were arrived at in good faith, but presented in a fashion that was too definitive.

The Committee further concluded that it was probable that:

four shots were fired
the fourth shot came from a second assassin located on the grassy knoll, but missed. The HSCA concluded the existence and location of this alleged fourth shot based on the later discredited Dallas Police Department Dictabelt recording analysis.[19]

So basically,the Committee found that there probably was a conspiracy,but said conspiracy didn't involve any actual people.
In 1988 the Dictabelt recording evidence from the Dallas Patrolman's motorcycle was thoroughly discredited. This was often cited as the ironclad proof of a 4th shot,so once that was proven to be bullshit,we are back to 3 shots,one nut,and a small time mook like Ruby.