Although @znt has gotten the answer through pari, I think it may be instructive to outline my argument.
It all depends on the transition function of Higher Ramification Theory: for a finite extension $K\supset k$ of local fields, $\varphi^K_k$ is a function from $\Bbb R^{\ge0}$ to itself, polygonal and concave, in which, if the (finitely many) vertices are $\{(x_i,y_i)\}$, each $x_i$ is a lower number of a ramification break, and $y_i$ is the corresponding upper number. Perhaps the most useful property of the transition function is that it is functorial: if $L\supset K\supset k$, then $\varphi^L_k=\varphi^K_k\circ\varphi^L_K$.
Although you’ll see below that there are other methods of calculating $\varphi^K_k$, here’s a relatively painless way for totally ramified extensions, if you know a prime element $\pi$ of the integers of the big field. Let $F(X)\in k[X]$ be the minimal $k$-polynomial for $\pi$, and draw the Newton copolygon of $F(X+\pi)$.
The copolygon of $g(X)=\sum_na_nx^n$ is the intersection of all the lower halfplanes $\eta\le n\xi+v(a_n)$, where you may take $v$ to be the (additive) valuation for which $v(k^\times)=\Bbb Z$. One sees that the segments of the copolygon are in one-to-one correspondence with the vertices of the Newton polygon: there’s a segment along the line $\eta=n\xi+v(a_n)$ if and only if $(n,v(a_n))$ is a vertex of the polygon.
And you get the transition function from this process by stretching the boundary of the copolygon horizontally by a factor of $e^K_k$, the ramification index (equal to the degree since our extension is presumed totally ramified). That is, every point $(\xi,\eta)$ on the boundary of the copolygon gets moved to the point $(e^K_k\xi,\eta)$ on the graph of the transition function. The function achieved in this way is shifted one unit up and one to the right from that defined in most standard texts, like Corps Locaux of Serre. In all cases, the infinite segment in the graph has slope $1/e^K_k$, and in the case of wildly ramified extensions, all slopes are powers of $1/p$.
Now calculate the transition functions of $\Bbb Q_2(3^{1/8})$, $\Bbb Q_2(\sqrt3\,)$, and $\Bbb Q_2(\sqrt2\,)$, all as extensions of $\Bbb Q_2$.
I won’t go through the details, except to say that $(X+1)^8-3$ and $(X+1)^2-3$ are the Eisenstein polynomials for the first two of the fields above. You find that the transition functions are: $\varphi^{\Bbb Q_2(3^{1/8})}_{\Bbb Q_2}$ has vertices $(2,2)$, $(4,3)$, and $(8,4)$;
$\varphi^{\Bbb Q_2(\sqrt3\,)}_{\Bbb Q_2}$ has the single vertex at $(2,2)$, and $\varphi^{\Bbb Q_2(\sqrt3\,)}_{\Bbb Q_2}$ has the single vertex at $(3,3)$.
Even though the transition function of $\Bbb Q_2(\sqrt2,\sqrt3\,)$ over $\Bbb Q_2$ can be calculated directly knowing that a prime element is $\pi=1-(\sqrt3-1)\big/\sqrt2$, you can also get it from the transition functions of the subfields, using functoriality. The upper numbers ($\eta$-values) of the subfields must appear among the upper numbers of the whole extension. Since these numbers are $2$ and $3$, $\varphi^{\Bbb Q_2(\sqrt2,\sqrt3\,)}_{\Bbb Q_2}$ must have the vertices $(2,2)$ and $(4,3)$.
All the above is straightforward and easy. But I wanted the transition function of the extension $\Bbb Q_2(\sqrt2,3^{1/4})$ over $\Bbb Q_2(\sqrt2,\sqrt3\,)$. For that I needed a prime element, and I found
$$
\beta=\frac{\frac{3^{1/4}-1}{\pi}-1}{\pi^2}-1\,,
$$
for which the minimal polynomial over $\Bbb Q_2(\sqrt2,\sqrt3\,)$ is of form $X^2+u_1\pi X+u_2\pi$, for units $u_i$. The upshot is that the transition function has the single vertex $(2,2)$. Compose this function first, then the transition function $\varphi^{\Bbb Q(\sqrt2,\sqrt3\,)}_{\Bbb Q_2}$, and you see that the transition function for the degree-eight extension $\Bbb Q_2(\sqrt2,3^{1/4})$ over $\Bbb Q_2$ has two vertices $(2,2)$ and $(6,3)$. The slope ratio at the lefthand vertex is $4$ rather than $2$.
And that’s enough to get our conclusion that $\Bbb Q_2(\sqrt2,3^{1/8})$ is totally ramified of degree $16$ over $\Bbb Q_2$, since its transition function must have a vertex with height $4$, coming from the subfield $\Bbb Q_2(3^{1/8})$. So the full transition function has vertices $(2,2)$, $(6,3)$, and $(14,4)$.
