Hi, I don't think there is anything wrong with 'the whole first part', but 'the whole of the first part' is clearer and better for written English.

the 2nd option goes more smoothly

you will likely receive contradicting answers to your question from both native Speakers and non-native speakers. Also from teachers and community teachers or people who are not teachers on italki. This is a learning platform, to learn the formal ways, that are taught in schools and expected to be observed, in business, during exams, and generally when confronting formal situations in the language. To survive on the street you must be aware of the colloquial uses of words/sentences etc. You have been given the formal correct answer to your question by, Rosie Kloska. Articles in English are vital and neccassary in formal or examination situations, more especially for writing. This means you must learn to automatically and instinctively write THE/A when or how it is needed. "The whole of the first part takes the form of a monologue" <-- is the standard you should be looking for at all times. you can be aware that the second example sentence may be used or heard or seen frequently but you must use the first in your own (official)learning.

the 2nd option goes more smoothly

the 2nd option goes more smoothly