I'm aware that similar questions have been asked, but none of them appear relevant to my own question.
By definition and by simple geometric observations, $N,B,T$ are all mutually perpendicular unit vectors that satisfy the following, where $\kappa$ is the curvature of some curve and $\tau$ is its torsion.
$$\begin{align*} \\N\cdot B=B\cdot T&=N\cdot T=0 \\ T'&=\kappa N \\ B'&=-\tau N\ \\ \tau&=-N\cdot B' \\ \kappa&=T'\cdot N \end{align*}$$
Since they are all perpendicular to one another, and of unit length, $N\cdot N=B\cdot B=T\cdot T=1$, and $N\cdot B=B\cdot T=T\cdot N=0$, where I use $\cdot$ to denote the dot product. The derivative of zero is just zero, so:
$$\begin{align*}&0=(N\cdot B)'=N'\cdot B+B'\cdot N=N'\cdot B-\tau=0 \\&0=(N\cdot T)'=N'\cdot T+T'\cdot N=N'\cdot T+\kappa=0 \\\therefore\quad\tau&=N'\cdot B\implies\tau B=N'\cdot(B\cdot B)\implies N'=\tau B \\\therefore\quad\kappa&=-N'\cdot T\implies -\kappa T=N'\cdot(T\cdot T)\implies N'=-\kappa T \end{align*}$$
But the formulae would have me believe that instead $N'$ is not equal to either $\tau B$ or $-\kappa T$, but instead is equal to their sum. My workings would suggest that $\tau B-\kappa T=\mathbf{2}N'$, which is apparently wrong! Where did I make a mistake?
The error starts with the two conclusions with "$\therefore$". For example, it doesn't make sense to write $(N'\cdot B)\cdot B=N'\cdot (B\cdot B)$ because you're confused with scalar multiplication of a number with a vector, and the dot product between two vectors.
What you can do to fix the problem is to write $N'=aT+bN+cB$, and then use dot product to get $a,b,$ and $c$. Namely $$a=N'\cdot T=-\kappa, b=N'\cdot N=0,c=N'\cdot B=\tau,$$ hence $$N'=-\kappa T+\tau B.$$