How can I use the pagination from the index() method inside the search() method?

You could add an extra method to your controller, e.g. private function show_list(). Put the pagination logic in this new method.
Use the index() method to fetch all records (without search criteria), and the search() function to fetch records that meet the search criteria.

At the end of both functions (index and search):

PHP Code:

$this->show_list();