Yeah.. two fish in a half gallon, changed once a fortnight are -going- to get sick. No two ways about it. And on top of ammonia poisoning, there's chlorine poisoning from no conditioner. Don't get me wrong, I'm not bashing you here, as at least you're here asking questions and finding out info, and we all make mistakes as beginners. But that betta is going to be in pretty bad shape, IMO it's a miracle he's still kicking now. Lots of clean, conditioned water in his own tank by himself and fresh water every two days (every day might add to his distress right now) for a while will maybe pull him through. Maybe purchase some aquarium salt (it's special salt from the fish store, not table salt) in case he gets worse. Salt can help ease symptoms of exposure to toxins, there's stickies on using it as a medication here (it shouldn't be added to the tank all the time).
ETA: Also, a product called Stress Coat could -really- help him heal the damage from no conditioner. It's really worth the price.
If he thrives and doesn't get sicker - great! I really hope he does, but realistically, he's been through a lot so don't be surprised if he does get sicker - please do ask for help if he does.. lots of knowledgeable folks here.
That said, fish have miraculously lived through worse, and he seems like a tough little guy! Good on you for looking for solutions, I hope you stay to read all the great info here on betta care.
actually, if you let the water sit 24 hrs the chlorine evaporates.
Well, it used to in the old days, if the water was left in an open tub in natural sunlight. These days, however, many water suppliers are using 'chloramine' not 'chlorine', which doesn't dissolve that way, which is why conditioners are for the most part absolutely necessary.
ETA: It also doesn't get rid of harmful heavy metals, like conditioners do.
Sorry, you've lost me - what doesn't cause chlorine sickness? If you mean leaving the water out to age, no that doesn't cause it, and I wasn't saying it did... But if the OP's water has chloramine in it, and the fish's water has ammonia in it (which I dare say it does..) then chloramine poisoning is likely, unless she leaves the water out for over 2 weeks at a time - as chloramine is bonded to the water moreso than chlorine.
Spending a few bucks on a small bottle of conditioner like Prime, which knocks out chlorine, chloramines and heavy metals all in one go just seems a better option for the fish, right now.