I'm not sure what to suggest as there are no clear symptoms - it doesn't sound like parasites. An anti-bacterial may help if the one fish is still showing signs of pop-eye as an infection could also be the cause of the other's lack of appetite.
The big tank won't show any ammonia until there's a source of ammonia in it. The small one with fish is at a very dangerous level of ammonia and you need to take action on that immediately.
As the small tank is not cycled and the larger tank may potentially still have some bacteria in the media your fish will be better off in the big tank. Even if it has no bacteria left, the larger water volume will mean toxins accumulate more slowly than in the smaller one.
Much of the beneficial bacteria will have either become dormant or died off, but there may still be some that will spring back to life once there is an ammonia source to feed on. I suggest getting a bottle of household ammonia to test how viable the bacteria is. Use the calculator in our guide to fishless cycling to dose to 3ppm then test 24 hours later for ammonia and nitrite. If either is present, you don't have enough bacteria and should continue as per a fishless cycle. https://www.fishkeeping.co.uk/articles ... hless-cycling-article.htm
However, what is being used to filter the smaller tank? If that is not a cycled filter the fish are at risk and you should test that one for ammonia and nitrite. If they are present, and the filter in that tank is not a cycled filter you would be better off putting the fish in the large tank with the previously cycled filter. This will give them a larger volume of water to dilute the toxins and the chance that the filter will build up enough bacteria quite quickly.
It's best not to change water during treatment so I would do a large water change immediately before starting the course and then wait until the end before doing the next one. Restrict feeding while this is ongoing as if they eat less they poop less and so water quality won't deteriorate as fast.
As this is a small tank and will have very little stock in it, 5ppm is perhaps too high. I would do a water change and get the ammonia down to 2ppm and then cycle at that level (i.e. wait for it to drop to 0 and then top back up to 2ppm). To get from 5ppm to 2ppm in a 5 gallon you'll need to change about 3 gallons.
The product may help. Wait a week or so and see if you get a drop in ammonia. You won't see nitrite until there's a substantial reduction. If you haven't already done so, turn the temperature up to 28C (assuming a heated tank) as warmer water is more friendly to the beneficial bacteria you're trying to grow.
Re: Where are the nitrites!? - a cycling query
OK - Don't add any more ammonia until the level has dropped back to 0-1ppm. Check nitrate in the tank and in your tap water and see if there's a difference and post back. Also, test tank pH and let me know what it is, and whether your water is hard or soft. Your supplier's web site will tell you this. Look for a page about water quality and you should find you can enter your postcode and get a quality report which will tell you the hardness.