Sure thing. Each wire is colour coded, strip a substantial portion of the rubber coating over the bare metal to expose the metal threads, on both ends. Then twist them to their correct color, as sturdy as you can and insulate the join with the tape, make sure wires from other colors do not touch each other. Carefully connect it up, make sure it works.
Now use an entire roll of electrical tape and start wrapping them around the whole cable at the join and a good to 10 cm before and after the join, like a bandage, but very tightly, up and down. Not ideal but now if you take care not to stress the join, it should last for a useable time before you have to do it all over again.