It is very difficult! I worked at a rescue where there were several horses they simply refused to put to sleep because they were their favourites. It often took a lot of convincing and a bit of knowledge to get them to even call the vet to check on them. I think you should be sensitive to her and maybe try and get her to see that the horse is no longer happy and enjoying life like he should be and it would be best to let him go so she can remember him how he was rather than in pain and discomfort. It is often more upsetting to keep holding it off. Offer to be with her when she finally calls the vet and tell her you will stay with her as long as she needs.
So there is a lady at my yard who's horse is suffering badly with Cancer and he has for the last few years. The tumours are now open and he is in a lot of discomfort. The vets have told her there is no more they can do as his body is not working with the meds. She has been saying she is going to book him in for 2 months now but changing her mind. I know how hard it is to lose a horse but watching him in this pain is horrible. Everyone has tried to tell her but it seems like nothing is going in, also with the summer coming there will be a lot of flys and I fear he will get all infected. How would you handle this?