When Magento's API doesn't work properly...

Write your own "API":

1. app/code/local/Engineering/Extapi/controllers/ExtapiController.php

<?php

class Engineering_Extapi_ExtapiController extends Mage_Core_Controller_Front_Action
{

	protected function _initAction() {
		return $this;
	}

	public function indexAction() {
		$this->loadLayout();
	}

	public function getnumberofordersAction()
	{
		if($this->getRequest()->getParam('pass')=="123")
		{ //check password
			$query = "SELECT count(*) FROM `sales_flat_order`";

			$resource = Mage::getSingleton('core/resource');
			$readConnection = $resource->getConnection('core_read');
			$result = $readConnection->fetchOne($query);	

			echo $result;
		}
	}

	public function getnumberofcustomersAction() {
		if($this->getRequest()->getParam('pass')=="123")
		{ //check password
			$query = "SELECT count(DISTINCT email) FROM customer_entity";

			$resource = Mage::getSingleton('core/resource');
			$readConnection = $resource->getConnection('core_read');
			$result = $readConnection->fetchOne($query);		

			echo $result;
		}
	}

}

2. app/code/local/Engineering/Extapi/etc/config.xml

<?xml version="1.0"?>
<config>
    <modules>
        <Engineering_Extapi>
            <version>0.1.0</version>
        </Engineering_Extapi>
    </modules>

    <frontend>
        <routers>
			<extapi_router>
				<use>standard</use>
				<args>
					<module>Engineering_Extapi</module>
					<frontName>extapi</frontName> <!-- URI -->
				</args>
			</extapi_router>
        </routers>
    </frontend>
</config>

Now - Don't forget to create the app/etc/modules/Engineering_Extapi.xml file:

<?xml version="1.0"?>
<config>
    <modules>
	    <Engineering_Extapi>
	       <active>true</active>
               <codePool>local</codePool>
	    </Engineering_Extapi>
    </modules>
</config>
Now you can reach the actions via:

http://yoursite.com/extapi/getnumberoforders/?pass=123

http://yoursite.com/extapi/getnumberofcustomers/?pass=123