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Perl, 88 bytes

@_=/./g;a:{for(0..@_-2){@_[$_,$_+1]=@_[$_+1,$_],redo a if uc$_[$_]gt uc$_[$_+1]}}print@_

Just a simple Bubble Sort. Call with -n option to pass the text.

e.g:

echo "tThHiIsS" | perl -n sort2.pl

Output:

hHiIsStT