Mix and pack the snow with your boots until it is evenly wet and fairly smooth, then move on to the next section. The kids can speed up the job by loading the smaller ones on a toboggan and dragging it back and forth over the slushy snow. If they fall off, they will be wet and unhappy. Before long everyone will be very unhappy with you, but keep them at it. Twenty by forty is a good size, but even a town house backyard has enough space for a rink.
Mom can go in early with the baby to make hot chocolate.
One year, when just enough snow had fallen, a short thaw and rainfall had the snow completely saturated and the forecast was for a hard freeze that night. We packed and smoothed the base by driving the car back and forth over it. It made the best base we've ever had. Another year, when the Christmas holidays were nearly over and there still wasn't enough snow, I sprayed water a half an hour at a time to build thin layers of ice on the bare ground. It didn't make as good a rink, but it was better than no rink at all.