Other mistake: When Amy reaches down to get the key out of her shoe she literally only bends slightly. There is no way that she could've reached all the way down to her foot, or retrieve the key in the seconds before she gets it. (00:14:30)
Continuity mistake: When the lights go off after attaching the belt to the ceiling, Holt is interrupted before he finishes the word "winner", however in the flashback he says "winner of the 5th annual-" before being interrupted.
Answer: In reality, the Commissioner is a civilian administrator, appointed by the mayor of New York City. He (as of this writing, the role has always been filled by a man; the show gets that part right) is usually someone who has risen through the ranks of the NYPD, but upon becoming Commissioner, is no longer a sworn member of the force (the highest sworn rank is that of Chief). Since Commissioners are appointed, technically anyone can serve in the position, including Deputy Commissioners (who are appointed, in turn, by the Commissioner himself).
Update: On 1 January 2022, Keechant Sewell became the first ever female commissioner of the NYPD.