If I'm not mistaken, the old copper was nickel plated.
I have a quick and easy method for dealing with old, flaky insulation:
Strip a few inches of NM conductor, and push the empty insulation onto the old wire, as far as it will go, then re-strip. If the fit is tight, use insulation one gauge up.
Usually, the splices themselves will keep the sleeves in place, even up inside the sheath/armor. If need be, secure them with a wrap of tape at either the emerging end or the stripped end.
