Helene is a Greek name and being Greek I am quite familiar with the name. It means sun ray or shining light. Greek people refer to themselves as Helena's. Helene is a version of Eleanor and is the base name for a plethora of other names:
Elaine, Elanor, Eleanor, Eleanora, Eleanore, Elena, Elene, Eleni, Elenora, Elinor, Helen, Helena, Helene, Lean, Leane, Leonor, Leonora, Leonore, Maclain, Mariaelena, Marian, Mariana, Marianne, Marilena, Marilene, Marion, Maryan, Maryann, Maryanna
Helene Phonetic Pronunciation: HEH-len (English)http://inogolo.com/pronunciation/Helene or
Heh-leh-nee or Eleni eh-leh-nee (Greek pronounciation)
http://babynamesworld.parentsconnect.com/meaning_of_Helene.html
http://www.thinkbabynames.com/meaning/0/Helene
I see where you disagree with the pronounciation of Helen but Helen it is as you can check the links above.
The French version is heh-leen or lene
http://en.wikipedia.org/wiki/Helene_(name)
Naturally, I prefer the original Greek pronunciation, but it look as if you have quite a choice in how you wish to say the name Helene. There are also many nicknames for Helene such as Lena, Leni, lene, elen, and so on.
It is a nice classic name that is not worn out on the most popular names list over the past few decades.
Good Luck, I'm sure whatever you decide will be just perfect for your new addition.
P.S. I love the name Eileen and again it is not overused!