First look into the _fields.php in /database/ In the $aOrdersFields there is neither field "iPayment" nor "sPayment". No field for Payment nor Shipment. They are in $aOrdersExtFields.
Then, when you look into the listOrdersAdmin function in /core/orders-admin.php, you will see, that the array $this-aOrders is created with "generateCache" function. Looking at this function you see, that only one database will be read: DB_ORDERS. The DB_ORDERS_EXT with the needed informations will not be loaded.
You have 2 ways to solve it (without changing database structure): Extend the generateCache function with loading the data from DB_ORDERS_EXT (fast) or in the listOrdersAdmin add the query to DB_ORDERS_EXT (slowly)
I have some HTML skills. All I needed is the updated php function. You done it well. Works properly. I have just changed mPayment to iPayment, to have an ID instead of the name. Thank you :)