I don't really know what a "private secretary" is, so i can't really tell if the translation is good. Ο ιδιαίτερος γραμματέας or η ιδιαίτερη γραμματέας (or more commonly η ιδιαιτέρα) is actually a personal secretary, but the word to word translation means special secretary (because he/she has a special place). Ιδιαίτερος can have the following meanings: 1) Special (in the meaning of exceptional, notable), 2)That belongs to someone, is his and characterises him (πχ. Τα ιδιαίτερα χαρακτηριστικά μου= my special characteristics)
Is the η like a regular ε sound?
No, not at all (and it never was).
η is like "ee" in "beet", ε is like "e" in "bet".
(In Ancient Greek, the sounds were different from now, but still different from each other; η was like a long French è sound [ɛː], ε like a short French é sound [e].)
Is there a way to make it spirited (not sure if this is the correct word) ?
I'm pretty sure that's not the correct word. I have no idea what you're trying to ask.
Do you mean as in spiritus lenis / spiritus asper (smooth breathing / rough breathing)? The rough breathing as in ἡ hasn't been pronounced in Greek for centuries, and hasn't been written since the spelling reform of 1982 (when the diacritical marks finally caught up to the modern pronunciation).
Do you mean accented to show a stressed syllable? Then yes: both ή and έ exist.