I'd say first make sure test kit is not at fault - do replicate measurements if you haven't yet.
But given that nitrates up too you probably have a source of organics somewhere that has been slowly breaking down and is now overwhelming your biofiltration capacity.
If you have sand, debris buildup can cause this at around this time. See if there's a spot that food and fish poo is settling out in the sand and collecting, and that's your problem spot.
Assuming you have sand, for now, you can increase cleanup crew (nassarius particularly handy with sandbeds) and skim more aggressively. If you have an obvious site of detritus buildup, you can siphon the sand bed but selectively only in part of the suspect areas - if you try to siphon the entire sand bed you will throw all of that crud into the water and can spike the ammonia higher. Then try to solve this by redirecting or adding flow.
If you have filter socks, clean or swap out more frequently - whatever you're doing, cut the time in half.
Finally, some people swear by dosing in a little microbacter 7 in situations like this - brightwell makes one and I'm sure there are others. If you have extra bottled bac from tank cycling you could try that as well. Just keep in mind that any bacteria you add in to eat ammonia/nitrates have to be skimmed out when they bloom - otherwise when their nutrient source is consumed they will die back and decay in your tank, leaving you back where you started.
Good luck!