Counting zero-sum subspaces for the multiplicative inverse function

Let $q=2^n$ with $n\ge6$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a subspace $E$ of $\mathbb{F}_q$, define \[ S(E)=\sum_{x\in E\setminus\{0\}}x^{-1}.\] Let $N_{n,k}$ denote the number of $k$-dimensional $\mathbb{F}_2$-subspaces $E$ for which $S(E)=0$. We prove that, if $k\g