spoilers*
to be able to meet kunglao in conquest you hane to complete shen/wen(not sure which)'s wight lotus headband mission. go to him, outside of the hut over by the tall towerish building, after you complete night wolfs training. he'll give you the wight lotus headband. you can only do this b4 hotaru arrests you. if you dont, shen and wen will already be dead. once you are free and whatnot you go to the main headquaters of the WL and kung lao will be there. you must perform a 10 hit combo on him and then you will be told to locate kai in outworld. he'll give u raidens staff. then go back to kung and get coins. thats about it. no1 knows wat to do with the staff.

To me it is strange that Kung Lao does not have the Hat.
Why is that, if they could just put his MKDA model in the game?

I had not crossed paths with Kung Lao in Konquest mode and I had beat the game with Shujinko. I did not plan on restarting Konquest mode to actually run into him, but my memory card unfortunatley had all of its memory erased. So I guess I will go back into Konquest mode and try to run into him.

Technically, Kung Lao does have his DA. It's his alt.