If you try to login at https://orbit.theplanet.com/Login.aspx?url=/Default.aspx (use any username/password combination), you can see that the login credentials are sent as a
Just create a POST request with that as the body, and give it to LWP.
my $req = HTTP::Request->new(POST => $url);
$req->content_type('application/json');
$req->content($json);
my $ua = LWP::UserAgent->new; # You might want some options here
my $res = $ua->request($req);
# $res is an HTTP::Response, see the usual LWP docs.