It sounds like you what you've tried is sorting the fields for each logfile.log record, when what it sounds like to want to do is sort the records themselves.

To do that, you need to save them all and then sort them, something like this:

while (my $entry = <DB>) {
   chomp(my @fields = split(/\t/, $entry));
   push @records, \@fields;
}

@records = sort { $a->[1] cmp $b->[1] } @records;

for my $fields (@records) {
   print <<ENDHTML;
   <li><a href="#$fields->[0]">$fields->[0] ($fields->[1])</a></li>
ENDHTML
}
[download]

(untested, and making some guesses about what you want)