High Nitrites-at a loss

the filter in the tank, when was it last cleaned ?

your right, has me baffled as well
a higher ph will push more ammonia, this might make the bacteria work extra hard to keep up, ... and then the trickle down effect as the next stage in the nitrogen cycle has more than normal to work with.

a lower ph won't fix this, but may make things easier (easier being a relative term) - not something i'd recommend just changing as we'd all rather avoid stressing the fish

as for other suspects (other than a filter full of waste)
could check your prime
could check your test kit (no idea how to test a test kit other than getting a new test kit and seeing if it gives the same results)
check the water going into the tank (that might work to test your test kit - but i don't know much about ammonia & nitrites and such in city waters, ... other than chloramine is clorine and ammonia together, which is very toxic and goes through some even more toxic stages in the pipes as it gets to your taps.

(i am so grasping at air here)
have a look into what conditions are key to a betta wanting to lay eggs/build a next, ... might be a better indicator of what's going on in the tank as it sounds like your fish (your daughters fish) is very very happy. this also has me wondering if the test kit is accurate or not

could squishy your filter media in a bucket of water, ... just to squeeze out any excess detritus it may have collected (no chlorine or such toxins though)

sure will loose some bacteria, but most will be stuck to the filter media itself, so i'd guess it should be alright

only quickly for this point, what we are told about the nitrogen cycle is overly simplified, ... sure there is a cycle, but there's also far more bacteria types that can interact in that cycle, ... some will move things in the direction we are accustomed to, others may move it backwards ... over-all i wouldn't worry about this as it's possibilities and not likely realities.

back tor reality, ...
low ammonia, good
very high nitrites - has us baffled
high nitrates - good for plants, not for fish, but acceptable

different question, ... where are your nitrites coming from ?
typical cycle ammonia is produced and converted to nitrites.
your ammonia is acceptable.
so where are your nitrites coming from
would have me wondering in this moment, what else is going on, not accusing your daughter but is she putting anything in the tank though the night or otherwise giving your fish more to play with (i'm assuming a young girl)
... i dono, just random guesses.

i have heard there is a sweet spot on the pH scale for bacteria production for the nitrogen cycle
don't remember what it is, ...
but even saltwater tanks get cycles going with a higher pH than 8.2

lots of ideas, lots of guesses, ... otherwise, like yourself, it just doesn't make sense

maybe just start over. not a few bowl, not starting from scratch with new substrate & filter media, ... but give everything a cleaning (nothing that would kill any of the bacteria - so no chlorine or whatever is in your tap water), but there's something that's going from somewhere in the tank into the water