Isn\'t it possible to set the type of a parameter to an Enum? Like this:
private getRandomElementOfEnum(e : enum):string{ var length:number = Object.
Another possible option not mentioned above is using the actual values. This is however possible only when you know all the options. This, in my opinion is definitely better than any.
doSomething(a: string, b: 'this'|'can'|'work'): void { //do something }