Well I commented out the use strict.

(Though I would like to fix it and use strict. But let us move past this for now.)

No we won't, because the stricture warning gives you a clue about what's happening. You are using something as a hash ref which isn't, at line 42, which reads

open(LOGFILE, ">>$self->{logfile}") or die "Cannot open logfile";
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See the hash ref in there, which isn't? You are dying at that line with an unappropriate message. Stick the filename you are trying to open in there, and $! to get the reason why something might go wrong.

Rewriting your sub log_message

sub log_message {
        my $self=shift;
        warn "\$self = '$self'\n";
        open(LOGFILE, ">>$self->{logfile}") 
            or die "Cannot open logfile '$self->{logfile}': $!";
        my $time=getLogTime();
        print LOGFILE $time, " ";
        for (@_){
                print LOGFILE @_;
        }
}
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I get

$self = 'Entering sub main::do_something
'
Can't use string ("Entering sub main::do_something
") as a HASH ref while "strict refs" in use at LogMessages.pm line 43.
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You are invoking log_message as a function inside your handler sub created in sub _log

                *{ $sym } = sub {
                        log_message("Entering sub $pkg\:\:$name\n");
                        my @ret = $code->( @_ );
                        log_message("Leaving sub $pkg\:\:$name\n");
                        return @ret;
                };
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and perl is just doing what you tell it to do...

_($_=" "x(1<<5)."?\n".q·/)Oo.  G°\        /
                              /\_¯/(q    /
----------------------------  \__(m.====·.(_("always off the crowd"))."·
");sub _{s./.($e="'Itrs `mnsgdq Gdbj O`qkdq")=~y/"-y/#-z/;$e.e && print}