Well, the only thing I can offer as an answer at this point would be to use a hydrometer. Measure the specific density before and after. You could then be able to tell how much in sugars you have left.
I can't really think of any good way to tell what bacteria and yeasts will remain in your strained water kefir. I know there has to be some since it will continue to ferment. There really should be some sort of rough guideline that can tell you about how high your probiotic count is based on time or the change in specific density.
I asked a yogurt maker at the store (were I work the cooking school) and he said it would be to hard to work out as it depends on how good your cultures are and so on. Each batch could be so different. He said it likely that your talking a probiotic count in millions-billions per cup and you would have to have each batch tested individually. Which yogurt makers don't really do. Yogurt is not kefir but it is similar so it should be close if not more.
As far as other vitamins and nutrients. You should see some B and C vitamins from the folic and acorbate acids. You can increase the folic acid (B vitamins) by fermenting for longer periods of time but then you will also produce more acetic acid (vinegar). You should be able to get more acorbate acid from more sugar and/or fruit but if you add a lot more you may need to add more kefir grain so it doesn't have to ferment to long(and have a risk of acetic acid again). It seems that this would likely produce more ethanol as well so you would have a slightly higher alcohol content (i have seen some people say they have gotten close to 2% alcohol by volume, but I think most recipes give you in the .2-.5% range).
Any other nutrients seem to have to come from the water (like a mineral water), the sugar (raw sugars tend to have more trace minerals), and the dried fruit.
On a side note, aside from the complex carbohydrate in the sugar, kefir is going to need calcium, magnesium, and potassium to continue to thrive. I remember talking to someone a few years ago that said they let there grains ferment/breed in a mixture of eggshells, raisins and some other things I don't remember now and again to get the grains back into tip top shape. But she also sold the grains so I don't know how needed doing that actually is.
Not sure you will find this to be useful, as I think you might have been looking for something more concrete. And the only DIY test I could find was for vitamin C and it wouldn't even tell you how much, you could just run a comparison before and after and see if it is more. It just seems like there are to many variables involved and not many people seem to do just straight sugar water. Hopefully maybe this will be useful to other people reading the question for at least a starting point on possible adjustments they can make.
Best Answer
According to http://www.soya.be/make-miso.php
"If temperature is too high the koji culture could become inactive."
"The container should be place in a clean room with moderate temperatures (15°C – 25°C)."
Like yeast I think the only danger is that if temperatures get too hot it will stop working or living. Your best bet at keeping it cool is likely putting it in a dark cupboard or somewhere dark with some kind of insulation ideally to stop large temperature fluctuations and stop it getting too hot. You get foil insulated picnic bags for example. Or newspaper or foam would also help insulate.