Drywall – How to Drywall Around Wood Extending From Ceiling

Personally I would pull them down and drywall the entire thing. The edge of cabinets is going to not look very well when your done if you don't and will stick out like a sore thumb. It's easy enough to pull down the cabinets too.

Also I would remove the drywall from the edge by the wall too and mud the corner. It's going to make it cleaner looking in the end.

Using backer strips to hold on drywall patches is a standard repair technique. Often the screws holding the backer boards are inserted through sound plaster or plasterboard from the finished room side.

Your approach seems fairly sound, but I would add a few screws from the good side, being careful not to crack the plaster too much as you install. Since you seem to have real plaster, I would predrill the holes, although when screwing through drywall, it is not necessary.

Finally, are you making some arrangement for a real vent to the outside? Moisture vented to an attic space is simply wrong, leading to possible moisture damage and mold. Closing off the bad vent is a start, but not the full solution for the problem. As @JamesOlson points out in his comment, an openable window is adequate ventilation; but if there is none, some other exterior vent is needed.