What you need to do is cut the connector off the controller and then get hold of a pair of Deans ot similar connectors. If they're Deans, solder the male one to the thick black and red wires. Then you need some reasonably heavy duty wire for the battery. You can get 30amp wire from Maplins. You then solder the 30 amp wires to the female connector one end and fix the other ends to the terminals in that black kettle connector, making sure that they are the correct way round. Mine had to go the opposite of the plug markings.It's a good idea to solder them after twizzling them round the screws. That's it. I did exactly that last weekend to con next the same battery to my Cyclamatic. If you can't solder, or don't have someone nearby, I'll do it for you, but you'd have to send me the controller and black connector by post. Also, I've got loads of Deans, so if you need some, I'll send you some. just send me a personal message with your address.
One other thing. You'll now have a spare thin red (or orange) wire from when you cut off the controller connector. It needs to be connected to a 36v +ve. There's a switch on the throttle, so figure out which two wires go to the switch and solder one to the thin red wire and the other to the 36v +ve and then you can switch the controller on/off at the throttle.
If you have any problems figuring out the 6 throttle wires, post again and I'll tell you what's what. I can't do it now because I need to look at my bike.
One more thing. When you connect connectors, check that you have matching colours on each side. They often put the wrong pins in the wrong slots on the throttle, pedal sensor and hall sensor connectors.