This is not to say it is a truly solitary bird. I've seen many Kookaburras sitting beside each other in this state. It just tells you that there are no rivals nearby, and any close proximity birds are in the same family group.
As they ramp up the volume and get into the laugh they'll tilt their neck right back and bring their body back into an erect state. The mouth will finally visibly open.
Sadly my example bird wouldn't laugh when I was at the optimum angle of the previous photos. So you're probably wanting to see a laugh from a better angle. The next two photos from my 2009 visit give you this view, but are of a different bird (I presume... they were taken only 300 metres from each other).
Another thing that tends to happen when Kookaburras really start laughing (as opposed to the light call of their display posture) is that every other bird in the general area will gather to the laughing bird(s). Meaning you can get anywhere from two to eight birds (in my experience anyways... I could see a higher maximum, but have never seen it myself).
It is at this point you'll be able to tell if it is a single family group your watching, or rival tribes. If the birds all just sit and call, they're friendly to one another. If they start chasing each other or taking swoops at the calling bird, they are likely rival groups.