Should a President fail such an exam, to whom do those giving the exam report? I know there is a constitutional amendment providing a way for the cabinet? Congress? to remove the President if he is incompetent, but I doubt the current people in those posts would be willing to do so.
To the President. It would violate HIPAA laws to give them to anyone else, without his consent.
Exactly. Despite the fact that the President is a public official, he is also a private individual, and AIUI, being the President does not trump (apologies for the pun) his right to privacy regarding his medical information, as guaranteed by HIPAA.
The fact that previous presidents have been more forthcoming about their health (both the good and the bad, such as cancer treatments and illnesses) is a matter of custom and a choice to be (relatively) transparent to the public, rather than a law mandating that they do so.
Similarly, previous presidents made a tradition of publicly releasing their annual Federal income tax filings, in the spirit of transparency, but there is no law requiring it, and the current occupant of the White House has chosen to ignore that informal precedent.
An insane President must realize he is insane and resign or call for himself to be removed. That may be problematic.
Having cognitive problems does not make one insane. It means one has difficulties with mental processes such as memory, attention, decision-making, and problem-solving. One may be insane without having these problems, or have these problems without being insane, or have both.
Besides, if you call the President insane, it may give him a legal defense against any crimes he may have committed, i.e. not guilty by reason of insanity.
If the physician deemed a president sufficiently incompetent, I assume he would pass it on to next of kin, it would fall to next of kin (Melania?) to deal with whether the matter should be made public; and resultant fallout like whether she then, for example, becomes power of attorney for all his financials. 
At the very least, filing such a request for declaration of power of attorney would effectively make the news public. I assume in that situation the doctor would be obliged to testify, although presumably being a military doctor there would be chain of command issues too?
This is a very specific legal defense, which requires the defendant to be unaware they were committing a crime due to impairment at the time of the event. Having cognitive or psychological disorders is not in itself a good defense. It’s rarely pursued in cases where it may apply as it can be weakened by e.g. evidence of planning behaviors, and when it is attempted fails some 75% of the time.
TIL that the last Physician to the President who was not a military doctor was Dr. Burton Lee, who served George H.W. Bush and (briefly) Bill Clinton.
My pint being that if “a” President were insane.
Crazy, narcissists don’t ordinarily own up or maybe even know they are like that.
Realization you have a problem is part of the battle. So a person with dementia wouldn’t necessarily be able to know.
Either way, Trump won’t be saying it.
If he’s insane, who knows what the insane think? Really.
(Until they write the manifesto)
As we have seen , an insane POTUS doesn’t need an insanity defense in order to get away with crimes against the American people and the world. There is not legal defense required for a King.
This is quite true, and combined with HIPAA, seems to ensure that a President with full-on dementia could stay in office until the end of his or her term. The only thing that could intervene would be an invocation of the 25th Amendment by that President’s VP and others authorized in Section 4 to initiate the President’s removal:
If those people choose not to act, a dementia-riddled President can go on doing as much harm as he’s able.
(My bolding)