Took apart laptop now keyboard wont work?

The keyboard ane touchpad are usually attached by printed wiring on an acetate-like material. The "white thing" sounds liek a tab that reinforces the end of that to enable it to slot into the socket - you should get a good idea where it is supposed to plug it simply by seeing where the end rests with the keyboard in roughly the correct position. Be careful with those leads, though, since they do get brittle with age. Even delicate handling can break the acetate and sever the traces which is invariably an expensive repair job.