I don't understand why do you want to add the contact form in a cms page and not use the default contacts page.
Even if you manage to make the contact form appear you won't be able to send messages through it.
This happens because the action of the form is
<?php echo $this->getFormAction(); ?>. In your case this will be empty and you will submit the form to the CMS page in which you added the form. And the page does not handle the post.
In the default contacts page the action for the form is set in the controller (I don't know why is this, seams like a bad decision from Magento):
->setFormAction( Mage::getUrl('*/*/post') );
My recommendation is to stick to the default contact page and if you need something custom in the form just copy the
form.phtml file to your theme and change it to your needs.