This is from a work of fiction, not a grammar, the product of a storyteller, not a linguist. Do not take artistic expressions too literally. The characters are not conducting a census.
Linguists may debate what the definition of a word actually is (it's something in between a morpheme and a phrase), but in no case would you be calculating some sort of decimal value to determine whether something is a whole word or not. The and a half simply indicates that from the author's perspective, a contraction should be counted as being greater than two words, perhaps because the presence of the apostrophe calls attention to the clipped middle word. But that is just a creative way of thinking about them. The author could also write that the sun burns, but that doesn't mean that the word burn means undergoes nuclear fusion in Canada.
I'm answering the question, however, as it may be useful for English learners to be aware that and a half is frequently used metaphorically for situations like this, where something obviously cannot be fractional, but a strict count of wholes does not seem to express adequately the value of the collective. Half is the most common, as it is the most basic fraction. You may come across quarter from time to time to indicate something that is a smaller fraction, but again, it is not the point to assign a precise numerical valuation to something.
Consider as an example someone with two young children and a puppy dog. You would not expect her to say she has three kids (although this is actually becoming more common), as obviously a puppy is not the same as a human child, and does not receive anything close to an equal share of time, money, or affection. Nevertheless, the investment in the puppy is significant, and requires a major adjustment of her lifestyle. Thus, in informal situations, she might joke that she is caring for two and a half children; this establishes that she is not treating all three as exactly equal (as a more clinical expression like three dependents might) but also that the addition of the puppy is at least analogous to having another child in the household.
In a different but similar sense, consider the long-running U.S. television sitcom Two and a Half Men. The title is taken from the situation, in which a man and his 10-year old son move in with his hedonistic brother. The son is starting to come of age, and so more than a boy. Still, he is not a full-fledged man either. He is counted as a "half man." This may also allude to the idea of two and a half kids, a different expression referring to the average size of an American household.
