Mailing taking weeks to complete
How can we tell if it was just this one individual who didn't receive it until today? 


The Successful Deliveries Report (Mailings > Scheduled &amp; sent Mailings > Report > Successful Deliveries) should show you the time each person was sent their email. You can also look at the mailings tab on each recipient.

I tested this on a sandbox where the Send Scheduled Mailings  job was not active and set the Mailer Batch Limit = 1, and Mailer Throttle Time = 120000. 

I had to execute the job manually multiple times, once for each email to go out. Perhaps your job did not process fully and then finished the next time it was run.