Only joystick support. There's no support for gamepads. That's why only one of the sticks work. And the way it works is precisly the way it should work (up-down move character and left-right turn the character). The Quake 2 engine underneath Daikatana thinks that it's a joystick.

However you can use third party tools (such as "JoyToKey") to map the mouse and keyboard functions to the gamepad. Or use the tools the Logitech provides with their drivers. (But I doubt that the cheap "dumbpad" like the Dual Action is properly properly supported by the Logitech's own configurator.)

http://www.electracode.com/4/joy2key/JoyToKey%20English%20Version.htm

Also I need to warn you that playing Daikatana with a gamepad makes the game insanely difficult since you'll be unable to hit the targets with the speed and precision required to complete the game. It'll only frustrate you. Trust me when I say that it's a very bad idea.