Strict mode should be enabled for Typescript

[singingwolfboy]

Typescript has many different compiler options, and the --strict option is very useful for ensuring correctness. Without it, type errors like #21 often creep in.

We should enable strict mode in the tsconfig.json file, and fix all the problems that are revealed in the process. Note that it will be much easier to do this while this repository is still small; if we wait until this project grows in size, there will be many more errors to resolve in the future.