I liked the episode.
I think what Kara did was not quite out of character considering she was furious at the time; a lot has happened since season 2.
She kind of behaved like Clark did when he found out J'onn had kryptonite. I am not entirely sure she should count on Lena's loyalty considering that it is Kara who is Lena's friend.
But... it'd have been a shock to realise that Lena has been keeping Kryponite all this while. Coming on top of her Sam revelation, which she didn't reveal even to Kara (not just SG), Kara may have decided to check her out for once.
On the other hand Lena lied to both Alex and Supergirl this episode. Not just about the Kryptonite, but also about Sam not wanting Alex to know. Alex decided things were fair because she thought the point Lena made was good; that it was Sam's call. But, Sam's call was to tell Alex, and Lena didn't allow her. So...yeah, while Lena's points about DEO may be right, Lena has not been exactly trustworthy either. And, she did harbour a mass murderer on her own cognizance.
After doing the same thing last season with Rhea, and it backfiring so spectacularly, one would think she would have learned her lesson. Apparently not.
I am not particularly mad at her for keeping Kryptonite; I always thought she had it with her; but I am mad at her for lying to Alex and justifying to DEO when what she did eventually blew up in her face. With SG there, the Trinity went off with Sam. Without SG, Lena would have been killed too. It was her fault; I thought there wasn't enough acknowledgement from the writers (not Lena herself; she seemed sad but James propped her up) that what she did was wrong.
I don't think they did a bad job with Lena in this episode; but I think they have been doing a bad job with Lena in general; in setting her up as this conflicted character. She never seemed to do anything remotely bad until this episode, so how are you going to buy that she has been planning (or having a backup plan) against Supergirl this whole while?
A couple of scenes would have helped so much. It also would have explained (in more than the outing yourself is your choice) way, why Kara still wouldn't tell Lena that she was Supergirl.
In any case, I believe "I have something that could kill you*" trumps "I am a superhero who everyone in the world knows about" in the general scheme of keeping secrets.
And "Oh the friend you wanted to help actually wanted to tell you her secret" trumps "Being a member of a secret organisation, I cannot tell you who I work for because that is not how government agencies work". In both cases, Lena's secrets were positively life threatening and a risk to a lot of people (and in one case, she didn't keep secret, she lied).
Unlike, Alex's or Kara's. (However, I realise that one could argue that if only Kara had come clean with Lena, she wouldn't have chosen this way. Not too sure about Alex because she kept the DEO thing even from Kara.)
Anyways: Lena was using a lot of false equivalencies this episode.
James was terrible this episode in my opinion (and just when I was starting to like him). He should have straight out told Kara that he liked Lena and hence, he will not do what she asked. Instead, he was too scared to stand up to her, and then, took the easy way out; that of lying; spoiling Kara's and Lena's relationship in the process.
And he broke both Winn's and Kara's trust.
Once he had decided to come clean to Lena, he should have asked SG to come clean about the Kryptonite thing.
I don't know... If they are going with the Lena will turn dark next season (or the season after), they seem to be going in the right direction. But, they seem to have started a bit late for proper set up.
*Note: I don't think it is a good idea to just destroy all kryptonite everywhere. There should be some backup plan. But, I can see how a typically decent, kind and rational person can be a bit irrational when it comes to Kryptonite.
Oh.. I continue to like Katie McGrath in these episodes; and of course Odette is great. I wish the writers had done a better job in Season 2 building Lena up; would have been so much better than all those episodes of Lena being a truly good person who just needs to be rescued by Supergirl.