I've never heard that pronunciation of Ina before and it seems odd. I don't like it and it's not intuitive for me. I would have guessed that Ina was pronunced ee-na and Ena was a trying-to-be-cool variation. EnaHazel is awesome!
I completely understand the IE-nə/EE-nə issue. The only "experience" I have with Ina was through Ina Garten and InaMay Gaskin. Both pronounce their name IE-nə so that's what comes naturally for me.
Ena is more natural for me to say, honestly. However it is a pretty incomplete sounding name to me.EnaHazel & EnaHarriet are my favorites.InaPearl being my favorite out of the first list.