@MittensTheKittensthat sounds tough.
My DM has a fall pendant, they look for someone nearby to be a respondent, does she know any neighbours? Also I think it was because of the fall pendant but it may have been related to the carers, we needed to install an outdoor key box ( it’s completely useless mind as DF always locks the house from the inside with the key in the door, but there we go).
In order to speak to DPs doctor, their surgery needed me to get a for, signed by her which I emailed across. Give them a ring, DPs surgery was super helpful.
Good luck - sounds like your DM is at that difficult stage, where she isn’t capable of living independently but doesn’t need a care home and refuses help. I laughed at her refusing the velco slippers for old people- what age is she ?