We induct on $n$. The base case is trivial. Suppose our conclusion holds for $n-1$ and that we have a collection of cards which sum to $k\cdot n! = (n-1)!(nk)$. Now temporarily split every card labelled with an $n$ into $n$ cards labelled with a $1$, and color each such card black. Now our cards have labels in the set $\{1, 2, \cdots, n-1\}$, so we apply the induction hypothesis to get $nk$ piles, each with cards summing to $(n-1)!$. Let $a_i$ denote the number of black cards in the $i$th pile. We apply the Erdős–Ginzburg–Ziv theorem (any multiset of $2n - 1$ integers has a subset of size $n$ the sum of whose elements is a multiple of $n$) to obtain $n$ piles from our $nk$ such that the corresponding $a_i$ sum to $0$ mod $n$ (this is possible as long as $k \ge 2$). Now we recombine the black $1$s into $n$s to get a (legitimate) collection of cards summing to $n!$. Repeat for $n!(k-1)$.