I am going to leave this post published, but I do not recommend this method anymore. The connection is just too fragile. I'll post a better method later.
Many modern electronic device use flat plastic ribbon cables to connect one circuit board to another. These cables are constructed by applying conductive polymer traces to a plastic backing. They cannot be soldered, and are usually connected to the circuit board with a special connector that uses spring pressure to create an electrical contact. If the connector cannot be removed from the board or bought from a supplier, the following technique can be used to attached wires to the ribbon.
First, tape the ribbon cable securely to a work surface (paper-covered tabletop). Then, strip the ends of some small-gauge solid copper wires (I like to use wire-wrap wires). Tape each wire in place, so that the wire is securely positioned over a trace on the ribbon
Mix up a batch of silver, conductive epoxy. This stuff is pretty expensive, but I'm always surprised how little I use for each project, so the supply of it lasts a long time. Load the well-mixed epoxy into a syringe, and use a luer-lock blunt needle tip. The diameter will depend on the pitch of your ribbon cable, but I would say .020" is a good start. The luer-lock tip is important since the syringe needs to be squeezed quite hard to get the viscous epoxy out. Just imagine that needle tip suddenly coming loose, and the syringe squirting epoxy all over!
Most conductive epoxies have a pot life of about 5 or 10 minutes (depending on ambient temperature). All epoxy must be applied before before the pot life ends, or it will not stick well even though it seems like it might be okay when it is applied. Once the epoxy is applied, I usually use an oven to accelerate the curing. I did not in this case because I was afraid of melting the plastic ribbon cable. Instead, I waited overnight to make sure the epoxy had set.
Finally, I use a hot glue gun to completely cover the wiring and epoxy joints. This provides a surprising amount of strength, and allows the part to be hanlded carefully without any other mechanical support.