Your cycle seems to be moving along, today you have less ammonia, more nitrite and more nitrate. So.....the ammonia produced is being converted better, but it's not done yet. As long as you have any ammonia or nitrite readings you need to be cutting them down with water changes. With a daily 50% water change you will cut those values in half which is what you need to do for the fish.
And yes, if you have media from an established tank to throw in a filter, by all means do it. It will definitely help put a quick end to your mini-cycle.
If you clean all your filters on any tank all at the same time, it's pretty certain you'll encounter a mini-cycle from disturbing/destroying too much of the good bacteria (the bacteria that converts ammonia to nitrite then to the less toxic nitrates) at once. Always alternate weeks on cleaning filters on a tank.
The filter media would be from a bio wheel, could I just float it in the tank or would it not do any good.