2 more ideas.
You said you had a bill faxed to you, it may be possible to fax them back but from experience they don't respond well to faxes.
1) Is there a line along the very top of the fax with the originating fax number?
Does your fax machine have a log function and if so there may be an originating fax number on it?
2) To pay online go to the Endesa website:
Then click on "Clientes" on the left side
then click on "Acceso Directo a Clientes" to the right of the picture in the middle
then look on the left side menu under Servicios Express
click on "Pagos Online"
then click on the orange sign on the right called "Ir a Pagos Online"
this will open a new browser window.
You have a choice, I presume you are not a user of Linea Abierta so the choice is narrowed to 2 options, hopefully your fax will have a payment slip along the bottom with a barcode so choose the button labelled "Pagar Ahorra" under the option called "Numero del código de barras"
This will take you to a payment page where you will need to enter the numbers from your barcode
Keep trying, we'll fix it somehow.