That is impossible.
1. The html is broken (no form tags), which is why there is no default form, to be set as default, or for you to select (via $mech->form_name or $mech->form_number), and why $mech->click doesn't work. This is a bug in WWW::Mechanize, it should throw a useful error messages, example

use WWW::Mechanize;

$ua=WWW::Mechanize->new;
$ua->get('http://search.cpan.org');#got a form
warn $ua->forms;
warn $ua->submit;
$ua->get('http://cpan.org');#no forms here
warn $ua->forms;
warn $ua->submit; #dies here

__END__
HTML::Form=HASH(0x1d76ca0) at - line 6.
HTTP::Response=HASH(0x1d77114) at - line 7.
Warning: something's wrong at - line 9.
Can't call method "make_request" on an undefined value at D:/Perl/site
+/lib/WWW/Mechanize.pm line 1641.
[download]

2. Clicking the button calls javascript, which is not supported (see WWW::Mechanize::FAQ). Besides, the javascript only redirects to rebootinfo.cgi (and it does that with eval -- whoever wrote this should be shot).