How to keep strands of thin multi-strand electric wire together ?

I am fixing a CFL lamp that requires each of two wires to be threaded through separate tiny holes in socket, keep straight, with all strands of wire together and go through holes in a simple springy piece of metal that is supposed to hold them in, I guess.

After finally figuring out that this was the way to re-wire this lamp, I cannot strip those thin wires and still keep all the strands of the wires together to make a good contact with the metal clamp, mentioned above.

How to keep all the strands together, after stripping the wires will help me very much. I've tried twisting them back together, bending them over to form sort of doubling of wire, using super glue and simply pressure. Nothing worked for me.

Thanks in advance for answers.