The topic is too long for a forum post, but I'd refer you to any one of the many splicing books that cover this technique. Brian Toss is a popular author although I can't guarantee that he's covered it for this size wire. I'll describe what I remember from the ancient past, but I'm likely leaving something out. One other thing, with modern ropes you don't gain much from the wire part, so you might consider replacing the wire portion with rope and avoiding the splice completely.
You'll need a small, sharply pointed fid to lift the wires. I used something like a 'Swedish fid' that has a cupped point that gives you a 'tunnel' for the rope to slide thru instead of a standard 'point' fid where you have to slide the rope next to the fid. The commercial fids like that are generally too large for this size wire so I had made my own out of a brass sheet that I bent around a dowel.
Dip the cleanly cut end of the wire into epoxy (just 1/4" or so) and let it harden to cover the wire ends so it slips into the core without snagging. Tie a knot 6' or so up the rope, then slide the cover back and trim the core- I don't remember how much - maybe 8"? Then slip the wire about 18" into the core past the start of the splice - about 24" before you unlay the core. Tape the core tightly around the wire. If length is critical, this is exactly where you need to set it.
Then unbraid the core for about 8" and tape at the end of the unlayment tightly to hold it in place. I can't remember anymore about the tails - you may be supposed to remove some of the rope strands before the splice? But I don't think so - it would clearly compromise the splice - this is where an experts instructions would be worthwhile. BTW, if your rope has an unlaid core (like Sta-Set, for example) I don't know what to tell you. The same instructions may apply, but more likely there's something specific you need to do.
Now comes the tricky part. Group the unlaid core strands into three tails. You'll want to tuck each rope strand under TWO wire strands at a time, so since your wire (probably) has six strands this will work.
Use the tip of the fid to gently lift two wires. The hardest thing here is to NOT snag a strand off the inner core with the sharp tip of the fid. It has to be sharp to get between the small wires, but risks snagging, so be careful! A bad snag will protrude from the splice, break, etc. All bad stuff. If you've got some spare wire, perhaps a cutoff or the old wire, practice this step and adjust the sharpness of your fid until you can do it consistently. You'll be doing it A LOT and it helps if you can do it smoothly.
Once you have the wires lifted, slip one strand under the wires going WITH the lay. This is backward from a normal three-strand splice, so it feels weird at first, but that's the way I was taught. Continue lifting and tucking, pulling everything smooth and fair for three tucks. Keep the tucks even on the three strands and pushed back toward the start of the splice but you want the wire to not be deformed - just expanded to contain the rope strands.
Then trim 1/3 of the rope strands away and tuck again, repeat that again and cut the strands off at the rope surface. Hopefully the 8" unlayment will be sufficient for this - I can't remember the exact amount. Some people recommend whipping the tail of the splice but on this small diameter it won't work very well. Tape over it if you want to.
Then skin the cover back over your splice and unlay it back to the end of the core splice and repeat the whole process with those strands.
If you get it right, you'll have a beautifully tapered splice where the wire strands show between smoothly flowing tucks of rope fibers. If you've got a lumpy, misshapen mess you'll know better next time!
Good luck!
Marty
