Replacing an old dart board ribbon connector - need advice
I have ordered the cable like that just yesterday. Search for "FFC" (flexible flat cable). Or "FFC 1mm" for example, if that is your pitch. There are connectors and adapters too.

You probably won't find 19 pins though, it's non-standard. But if your connector is OK you should be able to use it with 20 conductors cable.

So, your repair can look like this: Old cable cut above damage > clamped into 20p standard socket > soldered to old 19 pin connector.
Or, if the remaining undamaged part is too short, you can insert a combination of connectors and patch piece of new 20 conductors FFC in between.