In addition the big picture issues moritz talks about, here are a couple of little things to think about:

@theseWords = keys %Line;
@theseWords = sort @theseWords;
foreach $word ( @theseWords ) {
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You don't have to create a temporary array for each step here, you can just write:

foreach my $word ( sort keys @theseWords) {
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Also, it looks like you've slurped the whole file into an array of lines at the beginning, and for large files you'd probably be better off just processing directly line by line.