One first thing to note is that the <IN> in print $client <IN>; is executing in list context, which means you'll only ever call the $client->connected once, before you're sending the entire content to the client...

Writing the content line-wise might improve the situation somewhat

    while ( my $line = <IN> ) {
        if ( $client->connected ) {
            print $client $line;
        }
    }
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however, this is of course not yet a real satisfactory solution to your problem of the client (potentially) disconnecting before you can complete writing the line.

Update: BTW, some time ago, I was having a somewhat related problem (involving a pipe, not a socket, though...), in which I learned that such things can be trickier than I first thought... ;)   Maybe that discussion is of interest here, too.