600 calories a day doesn't put you in ketosis, it's what you eat, not the number of calories, Atkins isnt low calorie but you go into ketosis on that. Your ketostix may show you're in ketosis but they're not always accurate, you need to go by your body and if you're feeling hungry that's not normal in ketosis. You may be one of those people who's ketosis isnt affected by citric acid, but honestly having stuff that you're not supposed to on this diet probably isnt helping the mindset.
As for putting on 2lbs during the week, I dont think drinking during the night will help that as usuallly we weigh first thing in the morning as we've had a good long stretch of an empty stomach and if you're drinking all night your tummy wont be completely empty. Also that will affect your sleep pattern. I've noticed in myself and it's a well known fact that lack of a good night's sleep can affect your weight loss. If you keep waking up during the night and you're drinking each time you might find that is part of the reason.
There are other reasons why your weight might show an increase for a bit. Water retention is usually the major reason, usually because you're due on. Also if you've been exercising a lot in your first couple of weeks, your body will then go into starvation mode instead of burning fat. Is your stomach working? Being constipated can show on the scales.
I think first off you need to stop with the diet coke and change to coke zero which is allowed. Also stop with the sparkling flavoured waters you've found and swap to something on the allowed list
http://www.minimins.com/exante/197219-what-can-you-drink-exante.html. How much plain water are you drinking? That's the key to weight loss on this diet as it fills you up, stops you being hungry and flushes out the ketones etc. Have a week of nothing that's not 100% allowed and lots of plain water and see if that makes a difference on the scales. If it really doesn't make any difference to you personally then you can go back to the things you were having before, but I'd try it.