"I've done half of it" is okay.
Where do you want to put "the"? I've done the half of it? No, don't say that.
We do have an expression "You don't know the half of it!" which means "There is so much more to say about this."
You can also say something like "The half of the pizza with green peppers must be yours. You know I don't eat those."
In those cases, "half" is a noun, not a quantifier.
Student or Learner