I like thinking of documentation in layers.
At the top, you have requirements. That's your system bible. It is a high level description of what the system is expected to do. Everything "the client" expects must be documented there. That may sound obvious, but the system must implement what's in the requirements document and nothing else. In doubt on a behavior, you should always refer to the requirements. If a requirement is not clear or ambiguous, the requirement must be reviewed with the client before you change the system.
Then you have technical specifications, the system blueprint. How are you going to implement each requirements, or groups of requirements? Here you can think of class diagrams, etc. You can think of this as a set of technical requirements used to satisfy the client requirements.
At the very bottom of the documentation stack, you have code comments. Requirements are known and technical specs have been defined. Code comments must explain how a class fits in the big picture. If you have complicated algorithms, variable comments and comments inserted between statements help understand how they work and why they are used. Follow javadoc or doxygen conventions.
In the end, the documentation stacks contains a lot of well organized information about your system. Everyone working on the project will have access to the same basic knowledge.
Some people just hate writing. They won't like any type of documentation. I think you should simply have them agree on the necessity and usefulness of documentation.
- Docs are good to make sure you are implementing what the customer wants.
- Docs are good to share knowledge between every member of a team: devs, testers, managers.
- Docs ensure all members of a team have a common understanding.
- Docs are complementary to a training. It's better to have both than have none.
Do not document for the sake of documenting. Keep things simple and well targeted. I am not convinced a wiki is a good idea. I personally always worked with Word and I wouldn't change that. It is still the best tool to quickly edit, copy, paste text and images together from various sources, fast.