Finding the identity with permutation chains
Just a note: if the $K$ permutations can be arbitrary, and if you want to find the shortest possible solution, a quick reduction from the Pancake flipping problem shows that finding the shortest possible sequence is NP-hard.

The reduction: Given an instance of the Pancake flipping (i.e. a permutation of $[1..n]$), add to the list the $K = n$ possible prefix permutations.
The application of a $P_i$ corresponds to a pancake flip, and $U$ of them can generate the identity permutation if and only if the pancake of the original problem can be sorted using $U$ flips.

However, this doesn't apply directly to the problem as you stated it.  If we don't need the shortest possible solution, it's easy to find a solution of length $O(n)$ to the pancake flipping problem.  Since this question asked for any solution (rather than the shortest possible solution), there's no proof that this problem is NP-hard.