What’s the pH in the water you’re using? Also, what temperature is the tank being kept at? The efficiency of the nitrifiers is affected by pH and temperature - they prefer their water to be fairly warm, in the mid eighties Fahrenheit or even 90F if an heater will push it that high, just while you cycle. They grow fastest if the pH is around 8.2 or 3. They grow a lot more slowly if pH is closer to neutral or acidic or the water is much colder.
If your daughter has a cycled sponge filter, I would take it and rinse it out vigorously in the least possible amount of conditioned new water and don’t be gentle with it. Wring it, twist it, give it some hard whacks on the side of the bucket or container you’re rinsing it in - the idea is to knock some of the nitrifiers off. You won’t knock off enough to affect your daughter’s cycle but you will knock off enough to give your cycle a kick start. Pour it into the filter grunge and all. It may cloud water, if so, that’ll be temporary. If it spits some grunge out that settles on the substrate, you can vacuum or siphon that up later.
Way back when I got back into fish after a longish time away, I had to learn about the cycle too. It hadn’t been discovered when I first started fish keeping in my early teens.
My first tank was the same as the one I had way back when, 29G and I had two danios to cycle with and used Eco Complete substrate which was supposed to have ‘beneficial bacteria’ in it. Long story short, ammonia started to rise and then there was a nitrite spike. I was sure my fish would die but someone kindly helped out.
A fellow hobbyist gave me a bag full of what he called ‘filter rinsings’. He rinsed out the filter sponge from a filter on one of his tanks and gave me a bagful of rather gross looking water, which I simply poured into the tank. It made quite the mess on the substrate. But it also reduced both ammonia and nitrite to zero and gave me a nitrate reading in just 36 hours. The rinsings came from a filter on a fully stocked 60G tank - so there were plenty of bacteria in it.
So the filter rinsings can help even if they come from a small tank with a low bioload. The nitrifiers are relatively slow to reproduce, roughly once in 24 hours so it does take a little time but your cycle will progress even if you don’t kick start it with anything. But a kick start definitely speeds up the process.
