As KConrad points out, you perhaps mean to say that $F$ is a *finite* extension of $\mathbf{Q}_2$ or of $\mathbf{F}_2((x))$, and that the quadratic extesnions $E|F$ is *separable* (and hence galoisian) in the second case.

With this interpretation of the question, $N_{E|F}(E^\times)$ is a closed subgroup of index $2$ in $F^\times$, and every closed subgroup of index $2$ in $F^\times$ is of this form.  In particular, the ramification index $e_{E|F}$ does *not* determine the subgroup in question.

For more on this, see the relevant chapter in Serre's *Corps locaux* (=Local fields) or the book by Fesenko and Vostokov, among many other places.


**Addendum**.  In the same vein as David Speyer's example, it might also be instructive to work out the case $F=\mathbf{F}_2((x))$,by using "Artin-Schreier   theory" instead of "Kummer theory".  Cyclic quadratic extensions of $F$ correspond to $\mathbf{F}_2$-lines $D\subset F/\wp(F)$, where $\wp$ is the endomorhphism $t\mapsto t^2-t$ of the additive group of $F$.  The quotient $F/\wp(F)$ carries a natural filtration, coming from the filtration on the additive group $F$, so every line $D$ has a *level* in this filtration.  The ramification index of the cyclic quadratic extension $E_D=F(\wp^{-1}(D))$ of $F$ corresponding to $D$ depends only on this level.

It would be interesting to work out the norm subgroup $N_{E_D|F}(E_D^\times)\subset F^\times$ for each line $D$.

By the way, the [Second Extended Edition][1] of Fesenko-Vostokov is available online.

**Addendum 2**.  See [this answer][2] for a generalisation from $p=2$ to arbitrary primes $p$.


  [1]: http://www.maths.nott.ac.uk/personal/ibf/book/book.html
  [2]: https://mathoverflow.net/questions/84274/totally-ramified-p-extension-over-f-qx/84298#84298