PCG-71316L
But here's the problem, the model number isn't compatible with the laptop anymore because of the different Windows 7 Recovery Disc. What's more? We don't even know the disc's model number.
When they say the model # they are referring to the model # of the system itself, which should NEVER change. Short of replacing the system as a whole. This means if you purchased a "Supremo ABC123" system from XYZ manufacturer, reinstalling the OS will never affect this. You will always need to look on XYZ's support site for drivers for model # ABC123, etc.
The model # is normally found on the bottom of the system. Typically along with it is a serial# that identifies that unique system and is used for warrantied repairs.
PCG-71316L is not a valid model# for a current Vaio system. From what I have managed to get from the web, that model # is for a system about 7 years old. So its not surprising that Sony no longer has support for it. Many manufacturers limit support to X (usually 5) years.
If the model # you provided was from the disc from another system, then you gave the wrong info. As stated above, the model # needed is usually on the bottom of the system.
The model # is often also located in the BIOS of most systems.
Accessing the BIOS is different for every manufacturer and model.
Normally you should see a "Press (X) key to enter bios..." or "Press (x) key to enter setup..." when the computer starts.
It could be F1, F2, F3, Del, F10, F11, F12, etc..
The main page from the bios will normally have System, CPU, and memory information.
Write down the model # and then search it here:
Sony eSupport - Electronics - Select Your Model
If that still doesn't work, a last effort would be removing the wireless card and getting the information from it directly. This is not recommended as you can damage the system. A camera is highly recommended to use so that you can know exactly how the system looked during each step.
Essentially you would need to disconnect the battery & power cables first.
Secondly would need to ground yourself, else you risk damaging the system from ESD (electrostatic discharge).
Third would be careful removal and documentation of screw location and bottom panel. The wireless card usually has two or three antennae connectors and usually have 1-2(single/solid) wires connected to it. If needed, gently disconnect the wires and document where each wire connected. Most cards will have a connector similar in style to a memory socket but differing in the pins/contacts.
Here is an example of what a wireless card looks like:
http://www.irisvista.com/tech/laptops/Toshiba_Satellite_M65/big/Toshiba_Satellite_M65_04.jpg
On the card should be the manufacturer and card type if lucky.
Otherwise look for the chip on the wifi card, or at the minimum try to find an FCC code# on the card.
ALL PC parts are supposed to have an FCC code that can be used to identify it.
An example would be "Atheros AL503-BG2 V.3" or "Broadcom BG2321", etc.
Then google for drivers for that card type. Or look at the card manufacturer's website for drivers.