Okay, after much frustration, I figured out how to get Koethe to ask you to flip. You have to be at least Light I when you start the final quest. If you're not, he'll only wish you well and leave. That was the only thing I could think of that would make a difference, so I farmed LS points until my agent was Light I instead of neutral. This time, Koethe made the offer. So I was finally able to finish the fricking quest.
If you've already completed the quest, you can't change it. If you didn't turn it in, you can still reset the quest (provided you're not at the Star Chamber). Just make sure you get your agent to Light I. Hope this helps!