GB wrote

We have a very old Kenwood mixer, with aluminium beaters.

You sure ? It would be very unusual to use aluminium
for beaters, it isnt really strong enough for that.

We unfortunately put them in the dishwasher,
and they are now covered in a soft black deposit.

I did put some aluminium saucepans thru the dishwasher
in the past and they certainly end up quite different to
the way they were before they went in, but no soft black
deposit, just removal of the anodising.

What's the best way to get that off
and if possible restore the finish?

Hard to say without it being clear what it is.

Which detergent was used ?