So We killed Rikke and beat Tullius into submission. He's invincible, but heaving on the ground like an essential character that lost all his HP.

The two talk about how it should be my honor to kill him, but Ulfric never talks to me, Tullius just remains on the ground and no dialogue starts.

Any fix?

Have you tried letting Ulfric do the deed?

I misunderstood the quest. Check the edited body.

UESP has this:

http://www.uesp.net/wiki/Skyrim:Battle_for_Solitude#Bugs

General Tullius may also become invincible, as the scripted conversation that results in his death may not be triggered by beating him to the ground. Although the "Talk to General Tullius" dialogue option may still appear, the option cannot be activated, and the quest cannot be completed as a result.

• Striking Tullius with the http://www.uesp.net/wiki/Skyrim:Unrelenting_Force shout may fix this

This isn't a glitch I've run into myself, so I don't have any suggestions other than trying the above, and if that doesn't work, reloading a previous save from before you entered Castle Dour and confronted Rikke and Tullius. UESP does note that this scene can be particularly buggy so I usually make a save just before going through the door.

Even the divines give you sign.

Get back to the right path and away from ashtray way.