No one does. Frank Leslie's Illustrated News ceased publication in 1922. Under current US Copyright law all publications printed prior to 1923 now reside in the public domain.
Many earlier issues are in the public domain, but later ones are controlled by Financière L'Illustration.
I would assume Richard O'Brien still holds the copyright.
Unless other arrangements were made, the creator is the initial copyright holder.
Warner-Tamerlane.
Muppets, Inc.
Unless other arrangements were made, the creator is the initial copyright holder.
Catchme.com is registered to Slingshot Labs.
Little, Brown.
The Richmond Organization.
The Rank Organization.
You would have to have the permission of the artist or company that holds the copyright on the artwork.
Unless the job is work for hire or work for employer the photographer holds the copyright.