One possible reason for you to be having trouble finding the 'right' definition of Gauss-Kronecker curvature is that you haven't really told us what properties you want this curvature to have. I gather that you want it to be a scalar that is a function of the second fundamental form somehow and that, in the hypersurface case, it should be the classical Gauss-Kronecker curvature, which, in the hypersurface case, is defined as the ratio of the $\mathbf{N}_1$-pullback of the volume form on $S^{n-1}$ (where $\mathbf{N}_1:M\to S^{n-1}$ is the (oriented) Gauss map) to the induced volume form on $M$.
The natural way to generalize this is to generalize the Gauss map. In other words, for an oriented manifold and frame field as you have defined it above, consider the mapping
$$
^\ast\gamma = \mathbf{N}_1\wedge \mathbf{N}_2\wedge\cdots \wedge\mathbf{N}_{n-k}: M\to \mathrm{Gr}_{n-k}(\mathbb{R}^n)
$$
or, what, by duality, is essentially the same thing, the generalized Gauss mapping
$$
\gamma = \mathbf{e}_1\wedge \mathbf{e}_2\wedge\cdots \wedge\mathbf{e}_{k}: M\to \mathrm{Gr}_{k}(\mathbb{R}^n),
$$
where $\mathrm{Gr}_{p}(\mathbb{R}^n)$ means the Grassmannian of oriented $p$-planes in $\mathbb{R}^n$, which is a homogeneous space of $\mathrm{SO}(n)$ of dimension $p(n{-}p)$ and which carries a natural $\mathrm{SO}(n)$-invariant metric, $h$.
Then one natural definition of a scalar $S$ is to take the ratio of the volume form of $\gamma^*h$ to the induced volume form on $M$. Up to a sign, this reduces to the Gauss-Kronecker curvature when $k=n{-}1$ (i.e., the hypersurface case). The formula in terms of the coefficients of the second fundamental form is then
$$
S = \sqrt{\det(H_{j l})} \qquad\text{where}\qquad
H_{jl} = \sum_{\alpha, i} h_{\alpha i j}h_{\alpha i l}
$$
where the index $\alpha$ satisfies $1\le \alpha\le n{-}k$ and the latin indices satisfy
$1\le i,j\le k$.
Another natural scalar that is well-defined when $k=2$ or $k=n{-}2$ and $n$ is even is to note that $\mathrm{Gr}_{2}(\mathbb{R}^n)$ and $\mathrm{Gr}_{n-2}(\mathbb{R}^n)$ both carry a canonical $\mathrm{SO}(n)$-invariant $2$-form, $\Omega$. When $k=2$, you could just define a scalar $W$ to be the ratio of $\gamma^*\Omega$ to the induced area form on $M^2$. The formula is
$$
W = \sum_{\alpha} h_{\alpha11}h_{\alpha22}-{h_{\alpha12}}^2
$$
This agrees with the Gauss-Kronecker curvature when $n=3$ and generalizes it when $n>3$.
When $n=2m$ and $k=n{-}2=2m{-}2$, you could take $L$ to be the scalar that is the ratio of
$(\gamma^*\Omega)^{m-1}/((m{-}1)!)$ to the volume form on $M$. Of course, $W$ and $L$ are the same when $(k,n) = (2,4)$.
Of course, $S$, $W$, and $L$ generally have different properties, even when they are all defined. The good thing is that these things don't depend on a positive definite metric. They'll work for any submanifold on which the first fundamental form is nondegenerate.
I believe that the idea described by Ian Agol works, and can be elaborated on as follows. The general fact we want to establish is that the convex hull of a finite collection $X$ of points in $M$ is not smooth. Actually, the so called horro-hull, that is the intersection of all horro-balls which contain $X$ is not smooth. It follows that the convex hull is not smooth, because it lies in the horro-hull.
First let us recall that balls in $M$ are convex, because the distance function from a point in $M$ is convex. Given a ball $B$ in $M$ and a point $x$ on the boundary $\partial B$ of $B$, we can always find a larger ball $B'$ which contains $B$ and intersects $\partial B$ only at $x$. Letting the radius of $B'$ go to infinity yields a horro-ball which contains $X$ and whose boundary passes through $x$.
Let $B$ be the ball of smallest radius containing $X$. Then some point $x$ of $X$ lies on $\partial B$. Let $B'$ be a ball which contains $B$ and intersects $\partial B$ only at $x$. Then all points of $X$ other than $x$ lie in the interior of $B'$. It follows that every point in an open neighborhood of the center of $B'$ is the center of a ball which contains $X$ and whose boundary passes through $x$. Hence the normal cone of the horro-hull of $X$ at $x$ must have interior points, which yields that the tangent cone of the boundary of the horro-hull at $x$ cannot be flat.
Addendum (12/8/2023): More generally, if we have a finite collection of points in $M$ which are convexly independent, i.e., no point is contained in the convex hull of others, then every point is a singularity of the convex hull. Establishing this fact requires a bit more work, because in general there may not exist a sphere which passes through each of the points and contains the other points in the interior of the ball that it bounds.
To prove this, given a point $p$ from the collection $X$ of points, consider the convex hull $C$ of $X-\{p\}$. By assumption $p$ has distance $d>0$ from $C$. Let $C_d$ be the outer parallel body of $C$ at distance $d$. Then the boundary of $C_d$ forms a smooth ($C^{1,1}$) convex surface passing through $p$, while $X-\{p\}$ lies in the interior of $C_d$ (convexity of $C_d$ is due to the fact that the distance function from a convex set in a Cartan-Hadamard manifold is convex, and smoothness follows from the fact that a ball of radius $d$ rolls freely inside $C_d$).
Now note that by perturbing the elements of $X-\{p\}$ we may construct many different smooth convex surfaces which pass through $p$ and contain $X-\{p\}$.
All these surfaces contains the convex hull of $X$.
Hence as discussed earlier, the tangent cone of the boundary of the convex hull of $X$ at $p$ cannot be flat. Indeed the normal cone of the convex hull at $p$ (which is dual to the tangent cone of the convex hull) has full dimension.
Best Answer
Note that curvature in sectional directions tangent to the surface, say $\Sigma$ vanish. It seems sufficient to conclude that the usual Peterson–Codazzi equations hold for the surface. It follows that there is a convex surface $\Sigma'$ in the Euclidian space that is isometric to $\Sigma$; plus they have identical second fundamental forms.
Now we want to use bow lemma (diffgeometric analog of arm lemma). The bow lemma has a version for Hadamard spaces. Applying it to plane arcs in $\Sigma'$, we get that the map $\Sigma'\to\Sigma$ is distance-noncontracting.
Applying Reshetnyak majorization theorem, to plane sections of $\Sigma'$, we get that the map $\Sigma'\to\Sigma$ is distance-preserving.
Finally, applying Kirszbraun, we get that the map $\Sigma'\to\Sigma$ extends to a distance-preserving map from the convex hull of $\Sigma'$.