When I had to go to icu for a bad asthma attack I contacted my t so I can understand that. I don’t think it is inappropriate to contact her especially since you are close ( seeing her a long time, been through a great loss with her as witness etc.). Maybe a call would have gotten a response? My last would have seen me in the hospital if he had not been sick at the time. I can see how email could have been missed I’ve also had ya have email go into black hole of sorts as well. Talk to your t next time you meet and see what she says.