Lena finding out that Kara is Supergirl has been teased a lot - but I still don't see how that should make her a villain - or even just an enemy of Supergirl. Lena is a pretty smart, courageous character and to think that is all it would take to make her go rogue against her friends is pretty stupid. Kara as just Kara Danvers is still her best friend. Kara as Supergirl has saved Lena numerous times and they are on the same side. But somehow putting them both as the same person would make Lena hate her..? What..?
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I get it, they want drama in Kara's reveal, but please make it a short thing. If they need to make Lena get mad about it then fine. Have them butt heads a few times and then team up against the main villain, but let's not draw it out and finally put it to rest. They are friends, now leave them as friends!
(pssst...don't go full-evil with Lena, show, it won't work).
I am not sure what full evil is but it is easy to see how Lena could go fully antagonistic against Kara and Alex, at least for a while.
Lena is not a sadist, sociopath or psychopath. I think there is a very limited percentage of the population (probably 10%) that actually delight causing pain to people at all times.
However, the show has been pretty consistent at least as far back as back half of S3 (if not before) and especially this season in showing that Lena is driven first and foremost by her ego, her id. She has an extreme faith in her own talents and intellect, which is curiously combined with very poor judgment and extreme insecurity about her surname and her need to be taken at facevalue and not as a Luthor. She is easily manipulated by her need to be the best; and by her need for validation from her family (or parental figures); and her need to be seen as the Good Luthor.
She was manipulated in S2 by Rhea, S4 by Lex and Lillian... to singlehandedly cause a lot of Kara's problems. In S4: Kryptonite poisoning in the first half and Harun El in the second. Not to mention that the whole Children of Liberty thing was a result of Daxamite invasion, which was caused by Lena's poor judgement.
It is okay if someone has poor judgement; if that someone is willing to work with others and take advice from others. Lena doesn't do that. Even eve was just an assistant; all the decisions were made most of the time by Lena. She practically committed assisted suicide on a man who was not emotionally stable this season. Because, she didn't have the emotional bandwidth to consider the repercussions; and yes, had no checks on her by others.
She held a grudge against Supergirl (I know some might disagree on this) about the Kryptonite although Kara had tried to explain to her why she is so against Lena having Kryptonite. It was not about lack of trust in Lena, but what would happen in the wrong hands. (If Superman only trusts Batman with Kryptonite, Kara seems to only trust Alex and may be (?) J'onn.)
Ultimately, Kara had to apologise for Lena to agree to work with her.
She is also self-centred; I don't mean it in a bad way. I mean it in a kind of way that many people are. The entire back half of the season was about Lex destroying Supergirl's reputation and hurting her as far as possible and then aiming to kill her. However, if you look at how Lena reacts most of the time, it is an internal journey for her. When Supergirl is with her and goes with her to the prison, she doesn't think SG is there to work with her to find out what Lex is up to. She thinks SG is there to support her. In fact, for most of the Lex thing, her thinking is all about how Lex is manipulating her.
That he hates Kryptonians is an afterthought for her. Just like the fact that the Red Daughter is an actual person (instead of a weapon, an "it"; Kara had to correct her then) is an afterthought.
If she had accidentally found out from a stray footage that her brother had, she may have taken more positively to Kara and Supergirl being one and the same.
But the way Lex revealed it to her; playing to her ego; her fear of both being underestimated and being less than her family; her fear, lately, of being taken for a ride as Eve had done, the blow to her faith in her intellect... that would be very difficult for her to recover from on her own.
And, given the news, again, she chooses to do what she almost always does (though there was some initial growth this season that was then destroyed thanks to Lex and Eve). Instead of seeking help, she nurses the injury and the grievance.
She is right now at her most vulnerable, unable to think clearly, and because she can't bear the pain/the blow to her ego, most likely to blame Kara and may be Alex (because she has also been getting close to Alex).
It'd be very easy for someone to take advantage and lead her astray.
Say someone like the Leviathan and Eve. If they failed with one Luthor, why not the other?
They have Leviathan and J'onn's brother lined up as villains. They also have Lena. Why not combine at least two? I am not saying Lena will trust Eve; but they could make a deal of self-interest to destroy Supergirl between them.
I don't believe she will completely go psychopathic though. I think the writers and EPs love Lena too much for that; so may be by the end of the season (if is a 7-season show), she will be back in the fold. Brought back by Kara.
I think it is likely that Lena will not tell Kara that she knows. But, not sure how she will go after her. I thought may be through Alex, but not sure how that'd work either. And, Crisis complicates things.