That is where the problem lies, 'a person's ability to determine the nature of the illness truly helps a lot. Most people do not believe such an illness could be a demonic one.
So what do you suggest, when the person incharge does not believe that some of these people are under the torment of the devil?
I'll answer this with what I did. My friend and her son who lived in another state called and said her son was getting violently crazy even though he was on meds for his mental illness and she thought it might be a demon causing this. Her sister and a couple other ladies joined me in prayer to see if this was a demon and in fact it was. We started praying over this young man, but all we could hear was something laughing at us as we all heard it so I asked everyone to stop praying as we needed to regroup and be as one in the Lord as some got frightened and this was hindering our praying as we were not all one in the Spirit anymore and they had no business continuing as this is not something you want to play with.
I knew I needed to pray in private for the Holy Spirit to move in that room and in that young man before we started again so I went to get up and go outside to pray, but the door would not open and was locked on the outside as I tried to unlock it on the inside. I knew it was the demon trying to stop me so all I did was call out Jesus name and the door opened and I got into prayer with Jesus and he told me to go back in and start praying again and as soon as we came together again as one in Christ you could feel the power of the Holy Spirit moving in us and through that young man removing that demon from him. We knew it was gone and as soon as we were done praying and thanking Jesus the phone rang and all we could hear on the phone was it's gone, it's gone and my son is sleeping so peaceably right now.
I right all this to answer your question of what to do when the one in charge doesn't believe it's a demon.....pray anyway.
The effectual fervent prayer of a righteous man availeth much, James 5:16.