The crux of the matter is going to be how you would decide who qualified for your study, i.e., who was possessed. As designed, the study you propose requires all subjects to be possessed.
The Roman Catholic church describes the requirements for the diagnosis of demonic possession in The Roman Ritual
Especially, he should not believe too readily that a person is
possessed by an evil spirit; but he ought to ascertain the signs
by which a person possessed can be distinguished from one who is
suffering from some illness, especially one of a psychological
nature.[1] Signs of possession may be the following: ability to
speak with some facility in a strange tongue or to understand it
when spoken by another; the faculty of divulging future and
hidden events; display of powers which are beyond the subject's
age and natural condition; and various other indications which,
when taken together as a whole, build up the evidence.
The Roman Ritual, Part XIII, Chapter 1, para 3
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Other sects and religions have similar diagnostic criteria. In the past many religions believed that any form of disease was demonic possession. This has been superceded by advances in understanding medicine and pathology.
A second difficulty is going to be how you decide on the efficacy of your exorcism. Both groups would have to be held in exactly the same environment. For example, it's recommended that the victim be restrained during the exorcism and an exorcism can supposedly take weeks, even months. Any heroin addict restrained for weeks will have gone through "cold turkey" and the addiction will be broken. So, in theory, both groups would show 100% effectiveness of treatment.
The difference may be seen in the number who return to heroin from each group, but how do you determine what caused that? I'd suggest that going through cold turkey while someone yells passages from the Roman Ritual about demons would be enough to mentally scar even the toughest mind, demon or no demon.
Your case study would be better if you used something other that an addiction to mind-bending drugs, multiple sclerosis, cancer or HIV for example, but you'd be hard-pushed to find anyone that would perform an exorcism on anyone with a disease for which the pathology is understood. None of these conditions would fulfill the diagnostic criteria for demonic possession, unless you plan to consider all disease and illness as evidence of possession.
Having said all that, does demonic possession even exist?
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