I never seem to be able to complete the multi-step responsibility "eavesdrop on guild member" because I can't figure out how to "deliver secret war info to monarchy". It's happened every time I get this responsibility and I end up just taking the failure.

The info gets delivered to the Royal Advisor. It should come up with an orange map tag for them when this part of the responsibility is available. It tells you in the responsibility description to whisper the information to the advisor. Are you maybe doing the spy's responsibilty late at night when the advisor isn't available?

Nope. Same interaction seems broken for my jacobian priest. My advisor seems bugged.

Hmm....are you using the populous mod? because although that gives you 4 advisors, only one of them counts for the games quest and responsibility interactions.

Nope. My mod list is:

Flajko's PureEyes SmallTSMCompat
Grim's Core v2.0.13 beta
Grim's Female NudeMesh
Grim's Female NudeSkins
Grim's Male NudeMesh
Grim's MaleMaternityClothing
NRaas NoCD Medieval
Shimrod's NoRouteFailAnim
Rick''s spellfix
tummyza's TSM hairymaleskin test
tummyza's DynamicProfession

I don't use dynamic professions or nocd or spell fix, but unless you've added something to the advisor with dynamic professions or Grim's core, then I don't see any of these mods interfering with them.
You could maybe try resetting the sim (you can get the sims first and second name using grim's core) Also, is it just on this quest or has it been consistent through other quests. Have you tried the spy's responsibilities in a different kingdom?

Never tried a different kingdom but it does seem to be on all quests.

Then it does sound like your Royal Advisor might be bugged. If it's any consolation, random bugs don't travel across kingdoms so your next royal advisor should be fine.
Try resetting the Royal advisor and see if that helps.

I can't seem to kill her off because she's always associated with quests so do I just need to wait until I'm on a quest she's not involved in?

Save before you kill her. Reload if she doesn't get replaced or you can't finish the quest.