You somehow want to parametrise the vertices of the building of $G={\rm GL}(n,F)$ :
$$
G/F^\times K = BK/F^\times K= B(F)/B({\mathfrak o})Z(F)
$$
(by Iwasawa decomposition).

For $n=2$ ou can easily find representatives, but for $n>2$, it's going to be tricky!

I just give some hints. Write $N$ for the unipotent radical of $B$ and $T$ for the diagonal torus so that $B=T\ltimes N$.

-- If $n,n'\in N$ and $t, t'\in T$, then if $nt\sim nt'$ mod $B(O)Z(F)$, one has $t\sim t'$ mod $Z(F)T(O)$. So you may assume that $t$ is of the form

$$
t= {\rm diag}(\varpi^{k_1}, ...,\varpi^{k_n})
$$
where $(k_1 ,...,k_n )$ is well defined modulo the diagonal action of $\mathbb Z$ on ${\mathbb Z}^n$.

-- You have $nt \sim n't$ mod $Z(F)B(O)$ iff $n\sim n'$ mod $tN(0)t^{-1}$.

So for each $t$ as above, you need to find a system of representatives of
$$
N(F)/tN(O)t^{-1}
$$
For $n=2$, this is easy. For $n>2$, this seems tricky. I've never tried ...