Just compare the lowercase/uppercase variants of the strings. Make sure you convert all of them to common case (using the lc or uc methods). For example, I'd try a code like this:

foreach $orf (sort {lc($mips{$a}) cmp lc($mips{$b})} keys %mips) {
     push @mips, $mips{$orf};
}
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(note: didn't test the code snippet, but confident it should work)

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# Under Construction
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