I agree with the above answer. In a bit more detail, let E be a smooth conic in $\mathbb{C}^2$, of the form $xy=1$. Then the mirror given by Auroux's construction is $\widehat{\mathbb{C}^2}$ taking away some divisor as you say $D$. Well if we say that the point is (0,1) the blow up can be thought of as $ut_1=(z-1)t_2$ in $\mathbb{C}^2 \times P^1$. The relevant divisor $D$ is the union of $t_2=0$ and $z=0$. Thus we have something like

$$ uv=z-1 $$

where z is in $\mathbb{C}^*$. This is clearly the same as what we started with. In fact, these two spaces are even algebraically equivalent in this example, which would not be true if you consider some higher $m$, $A_m$ spaces. You can see this example studied in a lot of detail in the thesis work of Pascaleff.

All examples, following Auroux, Gross-Keel-Hacking, ... suggest that for surfaces, if the surface is "log Calabi-Yau," the slogan that hyperkahler rotation is mirror symmetry does seem to work (at least to some degree). Otherwise, it needs to be corrected along the lines that you suggest (superpotentials, birational transformations, deformations ...). There seems to be a higher dimensional analogue of this for higher dimensional hyperkahler manifolds, but there aren't as many computations in the literature as in the surface case.

I agree with the above answer. In a bit more detail, let E be a smooth conic in $\mathbb{C}^2$, of the form $xy=1$. Then the mirror given by Auroux's construction states that the mirror to $\mathbb{C}^2\setminus E$ is the complement of some divisor $D$ in $\widehat{\mathbb{C}^2}$. Well, if we say that the point is $(0,1)$, the blow up can be written explicitly as the zero locus of $ut_1=(z-1)t_2$ in $\mathbb{C}^2 \times P^1$. The relevant divisor $D$ is then the union of $t_2=0$ and $z=0$. Thus we have something like

$$ uv=z-1 $$

where z is in $\mathbb{C}^*$. This is clearly the same as what we started with. In fact, these two spaces are even algebraically equivalent in this example, which would not be true if you consider the analogous construction for $A_m$ surfaces for $m \geq 1$. You can see this example studied in a lot of detail in the thesis work of Pascaleff.

All examples, following Auroux, Gross-Keel-Hacking, ... suggest that for surfaces, if the surface is "log Calabi-Yau," the slogan that hyperkahler rotation is mirror symmetry does seem to work, at least to some extent (for log CY $A_m$ surfaces as above, the mirror is given by a hyperkahler rotation, in general the mirror will at least be diffeomorphic to the original variety). Otherwise, it needs to be corrected along the lines that you suggest (superpotentials, birational transformations, deformations ...). There seems to be an analogue of this for higher dimensional hyperkahler manifolds, but there aren't as many examples which have been explored in the literature.