The fastest time I have done "seeded fishless cycle" is five days. I manage to do this by placing an established filter (2 months running) in a new lightly planted tank.
If you need to use Hospital tank from day one, then use as much sponge media as you can from your fluval external filter and place it in the new fluval internal filter.
You may do better using 100% new treated water in the hospital, and be prepared for 50% partial water changes daily, until it cycles with your fish in it. Make sure you use a water conditioner such as Seachem's Prime, which besides neutralizing chlorine, chloramines and heavy metals, also locks up ammonia and nitrites for 24 hours (actually a little more than that) but keeps them available for beneficial bacteria to feed on.
Good luck and quick recovery for your fish.
Pepe
Santo Domingo