$$ \begin{array}{c|lcr} & \text{Player $\quad$ 2} \\ \hline & -2 & 8 \\ Player \quad 1 & 6 &-6 \\ \end{array} $$
$8$ us a pure nash equilibrium because it is where best response functions for each player intersect. And I have worked it out for mixed strategies.
$\{U, R\}$ is a pure strategy NE as is $\{D,L\}$. You can verify this by fixing one strategy at a time for the two players and looking at what each's best response is: assuming P1 players $U$, P2's BR is $R$, hence it gets one dot; it gets another because if P2 plays $R$, P1's BR is $U$. Any square with two dots after youve gone through all strategies is a pure strategy NE.