You do not give details on how large an area you got to back fill. You mentioned part of the wall under the driveway. Relatively speaking that is a small area. Too small to use a "vibrating plate compactor" !
Using that will unnaturally force your back fill downwards and then sideways - damaging your wall all over again! And it wont compact it properly because it does not magically remove air pockets and other crevices.
We had to do this allot when building on uneven places and do this under driveways too.
The best way to do it is layer by layer.
- Backfill layer of rubble (mixed with some ground is ok) (something like 30~60 cm)?
- Dump normal ground on top of it
- Hose it down with water. You should see the ground disappearing into the backfill, dump more ground until it is filled and overflowing. DO not flood it just get a reasonable stream of water. At this point the ground is naturally compacting(and getting rid of air space)
- When you reckong its pretty much not going to fill any more dump a thin layer of ground and damped it.
- Let it stand for while (15 minutes i n hot sun) and repeat from 1 till done.
This takes some time but will last for very very long time. When it rains, the ground is already water compacted so it wont subside. It also does not create shock damage to your retaining walls.