So cycling a tank is like when you first get a new one and you set it up but don't put the fish in until the water is 'right'.
Actually, cycling a tank is the process of growing bacteria. They need a source of food to multiply, in this case ammonia for the first species of bacteria. This comes from either a bottle of ammonia solution (fishless cycling) or from fish (fish-in cycling). If no fish are put in a brand new tank, then just letting it run does not cycle it because no fish = no food for the bacteria. But if ammonia solution is added to this empty new tank, there is food and the bacteria will grow.
Once the first species, the ammonia eating bacteria, start to multiply they turn the ammonia into nitrite. Nitrite is the food for the second species of bacteria and now they can start to multiply too.
It takes several weeks for both species to multiply enough to cope with the waste from a tankful of fish.
The majority of these bacteria live in the filter. It is in the dark and the water flowing through it brings a constant supply of food and oxygen. They live in the biofilm, which is attached to every surface in the tank - that white cloudiness you get on the glass and have to wipe away to see the fish properly is biofilm. The bacteria do grow in the biofilm on the glass, the decor etc, but not as many as in the filter.
This is why you have to be very careful about the stuff inside the filter (the media) as it is stuffed full of the good bacteria. If you throw it away, you have to grow a lot more bacteria. This is why we were all worried about you throwing away your old filter, complete with its media and bacteria. Because the tank has been running so long, there will be a good number of bacteria on the surfaces in the tank, just not nearly enough of them for the fish in there. But as Steve pointed out, getting the bacteria to start growing is the slow bit. You already have some, so it won't take as long to colonise the new filter as it does with a brand new tank.
You do need to monitor both ammonia and nitrite and do water changes as often as necessary to keep the levels below 0.25. Once they stay at zero by themselves, you can go to weekly maintenance water changes.
You do really need to look at your fish though. You say the plecs are fully grown - bristlenoses grow to 10 to 12.5cm. Zebra loaches need a tank 120cm long. Golden gouramis (a colour morph of the three spot gourami) need a tank at least 90cm long. None of these are suitable for a 65 litre tank, I'm afraid.
The cherry barbs and neons are fine though.