You can estimate $K$ using method of moments estimator
$$
\frac{k}{n} \approx \frac{K}{N} \implies \frac{N}{n} k
$$
or maximum likelihood estimator as described by Zhang (2009):
$$
\frac{N-1}{n} k
$$
for derivation and further details check the following paper:
Zhang, H. (2009). A note about maximum likelihood estimator in
hypergeometric distribution. Comunicaciones en Estadística, 2(2),
169-174.
On another hand, it you want to define a distribution of $k$ white balls, drawn without replacement from the urn containing $N$ balls in total, while treating the total number of white balls $K$ as unknown, i.e. as a random variable, then you can define such problem in terms of Bayesian model, with beta-binomial prior (in fact a conjugate prior) for $K$ (as described by Fink, 1997 and
Dyer and Pierce, 1993):
$$
k \sim \mathcal{H}(N,K,n) \\
K \sim \mathcal{BB}(N, \alpha, \beta)
$$
what follows to a beta-binomial posterior predictive distribution of $k$ parametrized by $N$, $\alpha' = \alpha + k$ and $\beta' = \beta + N-k$, and the posterior distribution of $K$ is
$$
f(K\mid k,N,\alpha,\beta) = {N-n \choose K-k}
\frac{\Gamma(\alpha+K)\,\Gamma(\beta+N-k)\,\Gamma(\alpha+\beta+n)}{\Gamma(\alpha+k)\,\Gamma(\beta+n-k)\,\Gamma(\alpha+\beta+N)}
$$
If you want to assume that $K$ can be anything in the $[k, N-n+k]$ range, you can use uniform $\alpha=\beta=1$ prior. For further details check:
Dyer, D. and Pierce, R.L. (1993). On the Choice of the Prior
Distribution in Hypergeometric Sampling. Communications in Statistics
- Theory and Methods, 22(8), 2125-2146.
You may also be interested in reading about capture-recapture method where you are interested in finding $N$, since it is closely related and follows the same logic.
