Hmm. As it so happens, I have a textbook on forensic psychology right on my desk that I've been reading, so I think I'll be able to field some answers/suggestions (paraphrasing from the book/my own opinions on the matter). Stuff that's taken from the book is "in double quotes", while italics are things I want to emphasize, and 'single quotes' are for...uh, terms? Anyhow.
For someone to be declared "Not guilty by reason of insanity" (NGRI), the judge/jury must decide, not the psychologist. This is an important distinction - although the psychologist's evidence/testimony can obviously sway the decision, it is ultimately left up to the courts to decide whether or not the person NGRI. The law states that someone cannot be held as criminally responsible if they do not hold mens rea - i.e., they did not have a "guilty mindset" while committing the crime. Legally, insanity is when a mental disorder causes the person who committed the crime to be free from 'mens rea' - they could not distinguish 'right' from 'wrong'. Thus, a direct answer to your question would be:
A defendant should be 'let off' (I will talk about this slightly later in the post) if the judge/jury finds that the defendant did not possess a guilty mind while committing the crime due to a mental disorder.
However, it must be known that NGRI is not actually a very popular defense - only 1-3% of all criminal cases (according to Golding et al., 1999 - something from the book) use this as a defense; acquittal rates range from 20-25%. It should be noted that (apparently) judges are more receptive towards the plea for insanity, whereas juries (supposedly) have a number of misconceptions/negative views on the plea (Callahan, Steadman, McGreevy, and Robbins, 1991).
In addition, one must note the difference between competency to stand trial and insanity; both are legal decisions (not clinical). Insanity was described above, but competency is the mental state of the defendant during the time of the trial, not during the time of the crime. Therefore, a lot of times, competency may be 'restored' through the use of drugs, which can make the defendant seem stoic and much more in control of their mind - although this may not have necessarily been the case (it might have been, as well) during the crime.
But how does a forensic psychologist determine someone's mental state? Well, in all cases, a psychologist will be looking backwards to try and 'figure out' someone...the book says, "in addition to a clinical interview, the evaluator should obtain copies of police reports, hospital records, statements of witnesses, any past psychological tests, and employment records, if possible" to try and give impartial evidence to the courts so that they are able to make a more well-informed decision.
With regards to defendants being 'let off' by an NGRI...I don't think that's a good way of putting it at all. A lot of times, those who are found NGRI are hospitalized to institutions, and according to Golding et. al., 1999 (again!) the individual is hospitalized for longer than they would have if they had served their term for being convicted. Note that you cannot be institutionalized immediately; there need to be all sorts of reviews/hearings to determine someone's mental soundness; someone also can't be kept for an indeterminate amount of time - there are periodic reviews and such. An individual is only released once "there is no longer evidence of mental illness...the exception may be the case of sexually violent predators..."
I think that this is a very interesting and complicated topic (albeit exaggerated by the media) that has many, many different facets. I don't know if what I wrote will help, but good luck on your essay!