Usually they fail because of inadequate strain relief, so that the wire breaks near one or the other connector. Thus, if you cut the cable in half, one of the two halves should be fine. You can then splice it with another half-cable to make the cable you need.
Mind : if you can make it work intermittently by flexing one of the connectors, you can cut it a few cm from that connector, & have nearly the whole length of the cable to work with.
That's the easy part, actually. You strip a little of one of the wires, use an ohmmeter to test between that & the corresponding pin on the connector (the colour code should tell you which one it is, but if it doesn't, you can just run the probe over the pins in sequence). If you get continuity, go on to the next wire. Eventually you should find one without continuity.
This shouldn't take more than 2 or 3 minutes per cable.
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