Correct me if I'm wrong, but this is what I did to complete the birthday.
1. When with Korra and her mom, choose the birthday over sex with Sinna (Dont remember her name)
2. Go to council, that prompts the ring and Jinora goes home. Choosing the sex with Korras mom over Jinora seemed to make Katara not mention the ring.
3. Talk to shady first, invite him to the party. Then just keep going to all the houses back and forth talking. I don't remember the correct order, just keep talking to everyone until you get the following to happen. (Talk to korra and she will help you get the ring in the MOUNTAIN not the mine, Korras mom bakes a cake, invite council, invite the twins, etc.) By the time this is over EVERYONE needs to go. Zhu-li will happen below.
4. Once everyone is invited tell Jinora to go to Korras, then tell Zhu-li to meet you there.
5. Go back to the Market to get the decorations before going to Korras
6. You know you get the scene because it autoplays after the party.
Thats it. Sorry this isn't super accurate. Its 4am and this is off memory. Let me know if this helps you. I'll clean up this little guide and get the order tomorrow if its not helpful.