Your LWP stuff looks fine, but you need to make a few changes to your regular expression syntax to change to make this work. Should get you started, but there's plenty of good documentation about using regular expressions about.
you need to use the pattern-match operator =~, not just a plain = to match a $string;
ie:
$string =~ m/
You need to capture something using the * modifier between the start and end of the comment you're looking for. A * just tells Perl to get as much 'something' (where something is the thing preceding the *) as possible. To get as much of anything as possible, use .*
ie: $string =~ m/($start)(.*)($end)/
The page that's coming back runs over lots of lines. By default, Perl only matches over one line to find a pattern. Use the /s modifier at the end of your regular expression to tell Perl to search over newline boundaries.
ie: $string =~ m/($start)(.*)($end)/s
Since the target webpage has got lots of pairs of matched comments, and because .* is 'greedy' (it takes as much stuff as possible while still matching a pattern), you'll get everything between the *first* $start and the *last* end. That's loads of stuff. Use the ? (non-greedy) modifier on .* to get just the first result item.
ie: $string =~ m/($start)(.*?)($end)/s
You don't need to put brackets around $start and $end, because in your programme, you already know what's in there. Brackets capture the stuff that's matched between them and save them, which you don't want to do.
ie: