Probability that all bins contain strictly more than one ball?

The probability is

$$\frac{K!}{K^N} \sum_i (-1)^i \binom{N}{N-i} \left\{ N-i \atop K - i \right\},$$ where $\left\{ n \atop k \right\}$ is a Stirling number of the second kind.

The $S_2(N,K)$ I have below satisfy $S_2(N,K) = K! \, T(N,K)$, where $T(n,k)$ is a 2-associated Stirling number of the second kind. (See also their OEIS entry.) The $r$-associated Stirling numbers of the second kind are the number of ways to partition a set of $n$ objects into $k$ subsets so that each subset contains at least $r$ objects. The subsets can be considered as indistinguishable urns, so to distinguish them we multiply by the number of ways to order them (i.e., $K!$) to get $S_2(N,K)$.

There is a known formula for the 2-associated Stirling numbers of the second kind. It's $$T(n,k) = \sum_i (-1)^i \binom{n}{n-i} \left\{ k-i \atop k - i \right\}.$$ See, for example, Fekete, "Two Notes on Notation, American Mathematical Monthly 101(8): 1994, pp. 771-778. (My apologies for the JSTOR link.)

Since the probability is $$\frac{S_2(N,K)}{K^N},$$ we get the result. So my comments below were overly pessimistic.

Original answer:

Charalambides's Enumerative Combinatorics, Exercise 9.23, says, "Let $S_r(n,k)$ be the number of distributions of $n$ distinguishable balls into $k$ distinguishable urns so that each urn contains at least $r$ balls." The OP is asking for $$\frac{S_2(N,K)}{K^N}.$$

For the $r=2$ case, the exercise asks to prove the generating function $$S_{k,2}(t) = \sum_{n=2k}^{\infty} S_2 (n,k) \frac{t^n}{n!} = \left(e^t-1-t\right)^k.$$

The exercise also gives the recurrence relation $$\begin{align} S_2(n+1,k) &= k \bigg( S_2(n,k) + n S_2 (n-1, k-1) \bigg), \:\: n \geq 2k, \\ S_2(n,k) &= 0, \:\: n < 2k, \\ S_2(2k,k) &= \frac{(2k)!}{2^k}. \end{align} $$

The fact that Charalambides does not include an explicit expression is not a good sign. The generating function and the recurrence relation may be the best we can hope for (especially since either differentiating the generating function $n$ times or unrolling the recurrence looks to be difficult).

Added: The $S_2(N,K)$ numbers are sequence A200091 in the OEIS. There's nothing there beyond what's in Exercise 9.23 in Charalambides's book.