Yes, you could add ammonia and monitor but some have successfully used a bunch of snails that they fed flake food to keep some bioload going that would feed the beneficial bacteria in the sponge filter.
Another thing you could do is to take a cup or two of gravel from a cycled tank and place that in the hospital tank to jump start the sponge filter - I assume that's how you're filtering the hospital tank - however it would take some time to get the bacterial colony going sufficiently but it does happen ultimately. Since the bacteria "anchor" so physical sights for the most part and there aren't appropriate concentrations in the water column, layering cycled gravel onto the sponge can speed up the process.
One other idea that can work is squeezing out a currently active sponge into the hospital tank. Live bacteria get squeezed out of the sponge and ultimately get pulled into the sponge filter of the hospital tank and begin colonizing the surface of the sponge filter.
