"Reunification of Skyrim" states "Regain Winterhold" and points me to Rikke. Unfortunately, she doesn't have anything to say about it.

I tried wiping out Fort Kastav on my own but that didn't help either.

I have no clue what to do.

You can work around this using the main quest.
When youre at the point you need to capture the dragon the Greybeards will hold a meeting negotiating a temporary truce.
At that meeting Winterhold can change hands.

Neat.

Thank you.

That's what I had to do. I just hope the quests don't mess up again after the mainquest or there will be no way to continue the civil war