Remove the clips in the housing that press to the top side of the ceiling.

If there is joists, holding strapping, holding drywall, cut two pieces of new strapping 16" long and place them perpendicular to the joists and parallel to each other on each side of the hole. You may have to wiggle them a bit to get them in place, but make sure that all of the ends of the new wood are under the nearest joists so that when you push up they will not go anywhere. Sand or plane them if necessary, don't force them under the joists because you might crack or deform the ceiling. If the hole is too close to an existing piece of strapping you may have to round out an edge of the new stuff. If the hole touches a piece of existing strapping then just use one piece of the new stuff and put it on the opposite side of the hole.

Pre-drill two holes opposite each other in the housing, a half inch from the bottom.

Using three quarter inch drywall screws, secure the housing to the strapping through the predrilled holes. You don't have to crank it in, the units with lamps only weigh about a pound, and you don't want to distort the unit. Just make sure it finishes nearly flush with the ceiling.

Spackle and paint, then install the trim. This is the hard part because the ceiling doesn't have a smooth finish.

Thank you for the advice, I'm sure my husband will enjoy working on this new project!