In short, no, it's not a Russian name.
It depends on what we consider Russian or non-Russian. Technically speaking there's very few names with Slavic roots in Russian. Let's limit list by female names and it would be: Надежда, Любовь, Вера, Владислава, Людмила and Светлана - and that's basically it. "Светлана" by the way has extremely interesting origin - it was coined by a poet in XIXth century.
Should we count Александра, Анастасия, Дарья, Мария, Ольга, Марина non-Russian then? Well, of course they are Russian names, since de-facto Russian parent use often this names.
What about Прасковья, Евдокия, Алевтина, Офелия - are they Russian? I mean nowadays it's not that easy to find Агриппина. However of course they are Russian.
What about Ilana specifically? It's a Hebrew name that's used predominantly in Jewish communities - to my knowledge not extremely popular but not exotic as well. There's no statistically significant (here's some open data on name distribution in Moscow if one is intrested) tradition in Russian or even Jewish Russian communities in Russia to give this name at birth.
This name, by the way, should not be confused with Ilona - a name of uncertain origin which is used predominantly in Hungary, Poland and and Czechia - but not in Russia.
Lastly, Лана suggested in the other answer provided actually is a short form of Светлана.
To conclude - this name is as Russian as, say, Роберта. Somebody can name their child Илана and the child honestly won't find any difficulties of bearing that name - yet another exotic name, not a big deal - Russia is multinational country where people got quite used to any exotic combinations. In fact, I've just realised that without googling I can name at least one Russian Ilana - Илана Юрьева from "Уральские пельмени" show.
As of the nickname part
I believe you are confusing nickname and diminutives, diminutive can be a nickname but nickname is not necessarily a diminutive. So the form for Ilana would be whatever Russian speaker will come up with - we are very productive in that aspect. A (definitely uncomplete) list would be - Ланчик, Ланусик, Иланчик, Иля, Илечка and so on an so on.