Searches for heavy neutral leptons $N$ in the $K^+\to e^+N$ and $K^+\to \mu^+N$ decay channels using the dataset collected by the NA62 experiment at CERN between 2016-2018 are presented. Upper limits on the corresponding mixing matrix elements are reported at the level of $\mathcal{O}(10^{-9})$ and $\mathcal{O}(10^{-8})$, respectively. This represents an improvement of at least an order of magnitude on previous measurement, as well as an extension to higher masses than previously covered. Related searches set upper limits on the rate of the $K^+\to\mu^+\nu\nu\bar{\nu}$ and $K^+\to\mu^+\nu X$ decay channels, with $X$ an invisible scalar or vector boson.