After many many different attempts, I finally got the files to attach. The steps that I took are things most people would normally try, but I added a twist to the first step:
1. Create a new blank database with the same name and schema: tables, constraint, etc. At this point you may have guessed what should follow. But before you go on, do the following: set it into emergency and single mode.
ALTER DATABASE CorruptedDB SET EMERGENCY
ALTER DATABASE CorruptedDB SET SINGLE_USER;
2. The rest of the steps are fairly typical. Stop the SQL Server service from the SQL Server Configuration Manager.
3. Replace the files for this blank database with the corrupted files.
4. Restart the SQL Server service from the SQL Server Configuration Manager.
This allowed me to access a set of database files that were otherwise no longer readable. Although I could not even look at the table definitions in the SQL Server Management Studio, I could selelct all the data rows out of it, and insert into a new database with the same schema: