while (my @names = $sth->fetchrow_array){
    $sth = $dbh->prepare("INSERT INTO names (name) VALUES (?)");
    $sth->execute($names[0]);
    $count++; #THIS IS LINE 27.
}
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You're clobbering your $sth. You have one to select (the while condition) and one to write (inside the while loop). Your selector is being overwritten, and replaced, by your writer. Try a new name for the inner $sth.

Update- you should also move the writer out of the loop. You only need to prepare it once and you can reuse it all you like.

my $insert_sth = $dbh->prepare("INSERT INTO names (name) VALUES (?)");
while (my @names = $sth->fetchrow_array){
    $insert_sth->execute($names[0]);
}
$sth->finish();
$insert_sth->finish();
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