You can't use a print statement in a runmode like this (better read the CGI::Application
docs more carefully: see the "Important note about run mode methods" section. It says, "never print to stdout"!). Change your "print" statement to this:
my $output = "Testing after login success.\n";
Then make sure your myrunmode.html template looks something like this:
<TMPL_IF OUTPUT><TMPL_VAR OUTPUT></TMPL_IF>
Note that the OUPUT variable in the template gets populated with the value of the $output variable in the runmode in the $template->param() statement.