TIMTOWTDI :-)
Here is my version, utilizing XML::Twig

#!/usr/bin/perl -w
use strict;
use warnings;

use XML::Twig;

my $t= XML::Twig->new(pretty_print=>'indented');

$t->parse( join('',<DATA>) );

$t->root->sort_children( sub{ 
                my $e = shift;
                return 0 if $e->tag eq 'header';
                return 10000000 if $e->tag eq 'footer';
                my @strength = $e->getElementsByTagName('strength') or
+ die "Malformed element:\n".$e->sprint;
                return $strength[0]->text;
        }, 
        type=>'numeric' );

$t->print;

__DATA__
<?xml version="1.0" encoding="UTF-8"?>
<system>
<header></header>
<class>
 <name>Eight</name>
 <strength>8</strength>
</class>
<class>
 <name>Four</name>
 <strength>4</strength>
</class>
<class>
 <name>Ten</name>
 <strength>10</strength>
</class>
<class>
 <name>One</name>
 <strength>1</strength>
</class>
<footer></footer>
</system>
[download]

The sort sub function is quite ugly and buggy,
if you are going to use Twig, you should either print the header, footer and the sorted classes in separate prints or surround all classes with a e.g. <allclasses> tag.

I just wanted to give an idea how you could use Twig for the job.