Table Sorting on "Content Indexed" View type
As indicated in the edit, the "Content Indexed" view type comes from the module Search API. After digging around I was able to find the page 

admin/config/search/search_api


Which houses a bunch of search indexing. For me, I was able to go into the "Content Index" and find a list of fields, with options for how they are indexed. At the top of this page there is the help text:


  The datatype of a field determines how it can be used for searching and filtering. Fields indexed with type "Fulltext" and multi-valued fields (marked with 1) cannot be used for sorting. 


I noted some the fields I was trying to sort on were listed ether as full texted, or did not have an option set for indexing at all. Setting these fields to something other than "Fulltext" allowed them to become sortable in the view.