Kyle kept saying things like "we're in a good place" and how they "worked so hard to get their friendship back" and she didn't want to jeopardize that. So, to me, that means she doesn't want tick her off and be on LVP's bad side again. If that means letting things like this go, then so be it. She probably has come to accept this as a flaw of Lisa's and a consequence of being her friend, but I think it also means she now walks on eggshells too. Was it Rinna who said Kyle's head is far up LVP's butt? I tend to agree with that.
Actually, no. They even showed in the flashback that LVP said she was "sorry for asking so many questions". I think Eileen is upset with her for the questions she was specifically asking (about the affair). So, LVP never acknowledged that her questions were too sensitive of a topic. She simply doesn't think it was bad to ask about it so she won't apologize, and Eileen will never be ok with that. I guess it could be construed as hypocritical of Eileen to be upset when she's talked about the topic herself, but LVP came off as nosey and probing with her question after question. That said, I probably would have let it go by now if I were Eileen.