Fridge Logic: Why would Kaede's mother have sent her a postcard, since she was likely to return home to her daughter long before the card made it to her? After all, she was just going to pick up Rin's parents from the airport.
In the anime it was made clear that Kaede's mom was going on the trip with Rin's parents, not just picking them up from the airport. She wanted to go with Kaede along, and presumably her husband and Rin, but Kaede was feverish at the time.
When Asa leaves the hospital, why does no one come running when the EKG flat lines?
Because Asa turned it off. It never had a chance to flat line. When you disconnect an EKG, you just get an error message. Since that can happen even when they're connected, nobody comes running when that happens.