It's worth noting that Henry V was not the first king to adopt English, as the phrasing implies. What Henry did was change the official language of court record-keeping to English, and he was the first king to use English instead of French in personal correspondence. Edward III was the first to promote the use of English at court with the 1362 Statute of Pleading, which made English the mandatory language of Parliamentary addresses and royal law courts, although court records were still kept in Latin until Henry's time.

In the 14th century English had been gaining more widespread acceptance among the higher classes; it had previously been considered a rough common language unfit for noble people to speak. This was no longer the situation in 1362, as most aristocrats had a working knowledge of it and the king could speak it. Poetry was being written in English. The first translation of the Bible into English would come only 20 years later in 1382. Edward's statute was enacted to solve the problem of citizens coming to court to plead their cases and not understanding anything said by their lawyers or judges.

I'm not an expert on linguistic history and others might want to speak up here; I just happened to be reading about this literally last night. My source is The Plantagenets: The Kings Who Made England by Dan Jones.