Get her to watch DS9 instead. Start with just Emissary, Battle Lines, Duet and In the Hands of the Prophets from S1.
For some reason, I couldn't make it past the first few seasons of DS9. Same with Voyager.
Did you try just watching those four episodes? And there are several in S2 you should skip, although the proportion of stinkers takes a nose dive and after all, TOS was about a third stinkers, too.
But you were right to bail on VOY.
I just shudder to see a newbie starting with TNG, considering that the first two seasons of TNG is hands down the worst run of
Star Trek of all time. And I don't think TNG has stood the test of time well, either. To me, it seems more dated in its mentality than the 2/3rds of TOS that are either passable or great.
Anyway, if you're determined to move ahead with TOS, here's how I'd shuffle things:
The Man Trap, Charlie X, Where No Man Has Gone Before - all worth watching, but save for later.
The Naked Time, The Enemy Within - start with these, they are high-concept with dramatic conflict that grabs the viewer quickly, and (crucial for the female viewer!) the conflict is very character-based, not based on a space ameoba or similar gimmickry.
The Corbormite Maneuver - save for later; space battles are a tougher sell for the female viewer.
The Menagerie, Pt I and II - show her these; more highly character based drama, based on personal loyalty.
Balance of Terror - I hate to say this, but save it for later.
Space Seed, A Taste of Armegeddon - show both of these; she should enjoy the recognizable elements from both that have entered pop culture.
The Devil in the Dark, COTEOF - show both.
If she gets this far, then you can show the other first season episodes and move on to S2. If you can hook her on the characters (and isn't that one of TOS' key strengths?) then she'll be more patient for space battles and technobabble stories and maybe not mind the jarring aesthetics so much.