The first thing to say is that the advice to water changes was good. You need to be doing them daily at the moment.
A quick lesson in the nitrogen cycle - don't worry, it's not too complicated

Fish make ammonia - it's in their waste and they breathe it out. But ammonia is toxic, it burns the fish's skin and gills making it hard for them to breathe. In a tank which has a cycled filter, there is a colony of bacteria in the filter which eat the ammonia and turn it into nitrite. But nitrite is also toxic - it binds to the fish's blood making it hard for the blood to absorb oxygen. In a cycled filter, there is a second colony of bacteria which eats nitrite and turns it into nitrate. This is only toxic at high levels.
Cycling is the term used for growing these bacteria and it takes several weeks, even months. Despite what the shop would have us believe, very few of the cycling products, such as the API one you used, do it instantly.
You are now doing what is called a fish-in cycle and it will need some work on your part to keep the fish alive until the filter cycles. This is something that very few shops will tell you.
The way to do a fish-in cycle is to do as many water changes of a big a volume as necessary to stop both ammonia and nitrite from ever getting higher than 0.25. If either are 0.25 when you measure, you need to do a water change immediately as it will go above that by the time you test again. To be honest, the lady at the aquatics centre is a bit cautious saying 25% every other day. You need to be guided by the test results; if necessary you can do 90% every day. Water changes are stressful for fish, but being poisoned is even more stressful.
The Pure Aqua may help or it might not. From now on, test the water every day, or twice a day if you can manage it, and do a water change every time you see the reading for ammonia or nitrite approaching 0.25. With just the danios in there they shouldn't go up too fast.
Here is a 'how to' thread on fish-in cycling
http://www.thinkfish.co.uk/forum/index.php/topic,17.0.htmlAs for the sickness, that is probably down to the poor water conditions. Having ammonia and/or nitrite in the water makes the fish stressed and more susceptible to infection. Keep on treating the fish. If the med has to be added daily, do a water change then add it. If the med is not dosed every day, on non-dose days add enough med to replace what was in the water you just took out eg if you remove 20 litres, add enough med to treat 20 litres.
Once you have successfully treated the fish, put some new carbon back in the filter to remove the left over med, then after a few days take it out and throw it away together with all the med stuck to it. Then, if it was my tank, I'd fill the carbon space with more sponge or ceramics as you don't actually need to run carbon full time.
Oh, and don't replace the sponges and ceramics like the manual probably says. You will have spent several weeks growing your bacteria, the last thing you want is to throw them away. Sponges and ceramic will last for years.