In this situation... Personally, I would do the following.
1 - Make sure you are only ever cleaning the filter media with old tank water.
2 - Daily 50% water changes - Using conditioned tap water
3 - Keep checking chemical levels till they are under control
4 - Keep feeding to the very minimum, and do it shortly before the water changes to ensure waste is rapidy removed
More filtration will be good in the long run, but is not going to really help much at this stage, as your filters do not contain enough biological content to deal with the bio-waste load. They need time to build up, and untreated tap water, too much waste is going to hurt growth.
Dont use chemicals to "lock away" ammonia, as it is required in some amount to be processed by the biological filtration.
The only quick fix here (in my eyes) is to bring in mature filter media from another established tank. This will give your tank a big boost in filtration and also help establish your existing filter media by seeding them, and reducing the ammonia load.
You can rectify this situation, it just needs a bit of time, and care.
Some people might disagree with me, but thats they way I would fix it
