@OP,
Hate to be the one to suggest regression, but how many games did you spend at King? I found the jump from king to immortal to be negligible; there was very little that I had to do differently and the results were about the same- maybe I finished games 10-12 turns later but nothing much else. Maybe you should have a few more games at King - try using civs towards victories they aren't designed for, like Genghis for culture, MariaI for dom, or Napoleon for anything other than culture. I found the big jumps to be from Prince to King (when you could no longer do whatever you want, have everything you want and began to need to prioritize/make sacrifices) and immiortal to deity(when many of the big boosts like wonders and wide empires were no longer an option and you needed to discover completely different boosts like tech stealing and trade route research).
Still, many of the aforementioned tips may help. Since the advancement you're asking about is king->emporer, I don't know if you'd benefit more from the prince->king tips or the immortal->deity tips, anyways:
-bring at least one worker with every land squad. 2 reasons for this are 1.)you can repair tiles in neutral or even enemy terrain in 2 turns and then pillage it for 25 HP whereas you'd only get 20 HP from 2 turns of wait-healing, plus you get some gold on top of that and 2.) lose him on purpose by leaving him unguarded 3 tiles out from enemy cities (you'll get him back the next turn anyway). this way when the enemy is lured out to capture him, you can finish the enemy without having your unit within 2 tiles of the enemy city, which means you'll lose the unit next turn.
-take note of which civ enters advancing eras first to find out who the tech leaders are. The civs with the highest scores aren't necessarily the highest in tech, and the civs with the highest numbers of techs may just be the ones who backfilled the tree. Then put your spies in that civs 2nd highest potential city, the highest potential is the capital but there's often a counter spy there, the second city takes a bit longer, but succeeds more than 95% of the time.
-immortal is the level when you can start to get a pretty decent amount of your research from trade routes. Internal routes are more-or-less still accepted as superior, but take a moment to see how many beakers you can get whenever it's time to renew a trade route. Additionally on this point, AI's will always send their trade routes to whatever city gives them the most gold and is in range. Use this knowledge to plan where EIC is going. Proper placement of EIC provides additional GPT, BPT, and can slightly reduce the chance of being DoWed upon. I'd argue it's the second best national wonder at high levels for this reason.