Thanks for the great detail! I've spotted exactly what occurred here. The "Long Term Care" mailing was scheduled to send to the "All members 7-16-15" group on July 28, but then that audience group was deleted from the Emma account after the point of scheduling the mailing but before the sending actually started. Here's a quick shot from the deleted items section of your account, which shows that particular audience group was deleted from the account on July 27:
When a mailing is scheduled, the system grabs the contacts that are in the group at the time of send to compile the campaign's actual audience. Because that group had been deleted prior to the mailing sending, there weren't any active contacts in the account in that group to send to. Hope this clears things up, but please give me a shout here if I can provide any further clarification. Always happy to help!