Is it possible to just add a new batch of apple juice and sugar to the sludge from a previous batch or does this increase the chance of infection and the chance of a failed batch?
Yes, I have done it myself on quite a few occasions, on the second use of the yeast the fermentation should kick in very quickly.
Bear in mind though, every time you do it though you increase the risk of infection and you really want to pitch the AJ as soon as the demijohn is empty.