Ok I read (quickly) the caresheet at Caudata Culture about Eastern Newts. I don't think it made me much smarter except that now I know that if your newt is an adult, it prefers to spend its time in the water. The only thing I worry about is that if your newt is an adult and normally in water, maybe he suffers somehow being on land? I don't know the answer to that, though, and you can certainly post a question about that in the newt forum.
Your photo helped me a lot more, though - and your tank looks very pretty, by the way. So, what I see there, I would treat as I would an axolotl (or fish) tank.
So. Cycling your water. It might be easier if you explain what you understand, then the others (where are the others just now, I am asking myself) don't have to spend the time typing out something you already know. In any case, the cycle works like this. A glass of water set on the counter will be fine for drinking (or living) even 100 years from now, assuming there was nothing in it to begin with and nothing comes into it later. If you put an animal into it, though, things start happening. The animal breathes in and out (oxygen etc) and it, well, poops. Amphibians also breathe (and release wastes) through their skin. We mainly detect this waste as ammonia. This brings us to the beginning of the famous "cycle".
Did you get all that about the cycle? The reason you need a "cycled" tank is that your newt produces waste (waste=ammonia), which is toxic, and the water needs to reach a (bacterial) balance to deal with that - like a pond.
Various sorts of bacteria "eat" ammonia and "excrete" nitrite, or nitrite-to-nitrate. And you want all of them in a balance that leaves your tank not like a toilet (full of ammonia and nitrite) but with only a little nitrate, which can be tolerated up to higher levels than ammonia and nitrite, but must be removed physically by water changes because nitrate is the end of the cycle, the end of what nature can do for your tank. (Live plants help remove nitrates, too, but we ain't even going to get into that now.)
So, you have had a "source of ammonia" in your water already. Maybe you put dead foods into the water or foods that died, maybe your newt went into the water to poopy. Whatever. Now you have ammonia, your cycle will begin (as it has). The thing is, the bacteria that change ammonia into nitrite will die if they do not get more "food". Food means "waste" - dead food, newt poopy. If the "ammonia bacteria" do not change ammonia into nitrite, the entire chain will fail.
So I am thinking, the best thing would be to get your newt back into the water. This means, like I wrote before, daily water changes of 20%.
Were you feeding your newt in the water before? If so and if you can convince him to get back into the water (by making the water more comfortable), then everyting should be fine - as long as you are consistent with the daily water changes. Ammonia and nitrite are truly deadly poisons, and so you just keep changing the water to keep their concentrations minimal while the bacteria are doing their eating-changing cycle.
I wish I could write shorter posts, but I hope this all makes sense?
-Eva