In this post on Power Drill grafting there’s a discussion (mostly between @RandallW and @Auburn) about grafting non-dormant scions (you’ll want to read the entire thread for context, but it’s not long.)
Short answer: clip all the leaves off of the wood you want to use as a scion, wrap the whole thing in parafilm or dip it in melted wax, and graft at once to the rootstock (healthy tree). Sooner is better.
Some people have had some success rooting hardwood cuttings, but apparently it can be tricky. Another possibility is that the tree itself might not be completely done- it’s amazing the abuse some of these things can come back from.