I have no idea what happened plot-wise because I spent this entire episode shrieking with glee and making clutching motions at Elijah's face.
THIS JUST IN: I still love Elijah for reasons I cannot for the life of me explain to rational humans.
I... I would like to believe in Katherine and Elijah because they're hot and awesome and smarter than everyone else and would be super great together but... I still think Katherine might actually be a shark with killer heels. I'm not sure she's capable of feelings like that anymore. (To be fair, she isn't sure either.)
ETA: (On a second viewing, I have had a slight change of opinion. To wit: Katerina Petrova is a badass bitch who has been running and manipulating people and kicking ass at it so long she doesn't remember how to do anything else. And I think she both has feelings for Elijah and finds his feelings for her useful in a two birds one stone sort of way. And it's kind of sad that she had to give up control to get him to sort of trust her because she's been doing great so far on her own, but also perfect because the only way he would trust her is if Katherine Pierce gave up control of something.)
(On the other hand, it occurs to me that with Elijah's love and protection, Katherine doesn't really need to bribe Klaus off her back. She's sort of the least of his problems at the moment and keeping Elijah's goodwill should be enough incentive for Klaus to leave her alone. Food for thought.)
Everyone needs to just fucking accept that Elena is a vampire. You have eternity to get her to turn her emotions back on. It will happen with time. You have plenty of it. Just stop. Let her run her own life.
Also, um, guys? Bonnie and Silas can literally start the apocalypse any day now. Maybe worry about Elena's emotional state later. Priorities.