Why don't ionic compounds show stereoisomerism?
They can, and they do.  The secret is to have a chiral cation and separately a chiral anion.  Letting $D^+$ be the dextrorotatory form of the cation, $L^+$ be the levorotatory form, and anologously for the anion, we then have four isomeric salts:

$D^+D^-$

$D^+L^-$

$L^+D^-$

$L^+L^-$

The first and second are diastereomers because only the anion is mirror-reflected; the cation is not mirror-reflected.  Similarly for three of the other five possible pairs; only two of the six possible pairs are enantiomeric.

Being diastereomers instead of enantiomers, the first and second salts, for instance, may have different physical properties such as solubility in a given solvent.  This may be used for separating different enantiomers of the anion (or, with a different pair such as first and third, separating different enantiomers of the cation).  And it's actually done.  See here for a brief summary and here for an example of diastereomeric salt formation.