Took a gamble on eBay and bought one of these cables. I did some research on how to find the right cable. After doing this, I picked opposite-side contacts, 0.5mm pitch, 10cm length and 50-pins. Cable came today and it fits the laptop perfectly. The laptop now powers on and everything seems to be working fine. Laptop does have other faults, but these existed before the cable broke (e.g. one of the keyboard buttons do not work).
Glad the cable worked out well. Saved me some time. I can now get back to troubleshooting this laptop further as it did have other problems. One of the cables breaking was an unfortunate roadblock.
While I'm here, there is one issue I'd really like to address. That is the battery. A third-party battery was installed on it a while ago and unfortunately the laptop won't accept it. What happens is when I first insert the battery and then connect the charger, the battery LED will light up orange. After maybe 30 seconds, it begins to rapidly blink orange. In this state, I can power on the laptop (even if charger is not connected). On Windows, it says it has 255% available and is plugged in (when connected to charger). If I disconnect the charger, it will stay on and just say "Battery 1: 0% and in use".
If I shutdown or power off the laptop, it will no longer turn on. This is with both battery+charger and battery+no charger. If I disconnect the battery and re-connect, it will turn on again and I can repeat the above.
Any idea what is going on here? Is this a battery fault, or an issue with the laptop? I've read online that some VAIO laptop block third-party batteries from being used. Could this be the case here, or is it just a bad battery?
The battery is a Green Cell VGP-BPS24.