After upgrading to 2.5: Table 'alpineductless-craft.craft_elementindexsettings' doesn't exist
As Lindsey D pointed out in the comment, the answer was to restore the database backup from the automatic backup located in /craft/storage/backups that is generated when the update was initiated.

When this backup is put into place it will re-run the database migration when you visit the Admin area.

So my full solution was 2 steps: 


Replace /craft/app with a fresh copy
Restore the automatically backed up database in /craft/storage/backups