We propose a novel estimator for the number of components (denoted by $M$) in a K-variate non-parametric finite mixture model, where the analyst has repeated observations of $K\geq2$ variables that are independent given a finitely supported unobserved variable. Under a mild assumption on the joint distribution of the observed and latent variables, we show that an integral operator $T$, that is identified from the data, has rank equal to $M$. Using this observation, and the fact that singular values are stable under perturbations, the estimator of $M$ that we propose is based on a thresholding rule which essentially counts the number of singular values of a consistent estimator of $T$ that are greater than a data-driven threshold. We prove that our estimator of $M$ is consistent, and establish non-asymptotic results which provide finite sample performance guarantees for our estimator. We present a Monte Carlo study which shows that our estimator performs well for samples of moderate size.