Estimating an unknown distribution from a mixture

I have two data sets, $\{x_i\}$ and $\{y_i\}$ . I know that data set $\{x_i\}$ was sampled from some distribution $X$ , and that data set $\{y_i\}$ is sampled from a mixture of the $X$ , and some other unknown distribution $Y$ . I am wanting to estimate what the mixing ratio is/to know how many of the samples in $\{y_i\}$ come from $X$ . If I make some assumptions about $Y$ (such as it being normal) this is just a simple mixture model problem, but ideally I don't want to do this. I'm wondering if there is some approach to this problem, or if it isn't possible. One idea that I had was to have a bunch of kernels (evenly spaced normal distributions with known $\sigma$ ), and use MLE to find their mixing ratios, but I assume doing so would just set the mixing ratio for $X$ to be zero, and just give me the KDE. Perhaps there is some way of penalising this, but my only thought was to set a prior on what I thought the mixing ratio of $X$ was, which I would rather avoid. If it is possible to solve this problem for categorical mixture models, than I can just bin my data, but I couldn't find a way of solving this problem in a categorical sense either, or really anything to do with parameter estimates for categorical mixture models (which makes sense because the sample distribution would have the maximum likelihood)