That's actually not as bad as I was thinking! At least the ammonia is 0, I expected that to be at least buffered by the plants for sure. Nitrites are high definitely, I think it's in the process of just establishing itself and Nitrates are really high!
Can you do at least a 50% water change? That will help to start. What I think should happen is, can you do live plants in the water as well? This will help to buffer at least from Ammonia and Nitrites, nitrates are normally eaten a little by plants and then mostly taken out by water change. But the toxic stuff will at least be buffered by plants. You could so like an Anubias and then some water wisteria or something fast growing to soak up the ammonia/nitrites and that should help!
I really do think this can work, just needs to the right things for it to happen ^_^ I definitely think a water change at least once a month is necessary since there's no filter to get the non-dissolved organic compounds out along with fish waste and stuff like that. But right now since it's in the process of establishing some BB on the walls and gravel and such, I think you need to test daily and do a small water change while it get's established. Just like you have to do with every tank and then once it's established, the once a month cleaning should be fine
But I think some low light, slow growing plants; Java Fern and Anubias, will definitely kick it off in the right direction ^_^ It will help improve the water quality and your little one will love to play/rest in them