I've mulled over this a fair few times before, and more and more the conclusions I start to draw are that the war will begin turning against Henry V before he can complete it. It's not so much the Joan of Arc factor - I like to believe that he would probably find a way of drawing her out and defeating her on the field of battle - but that the Dauphin's forces were shaken out of a stupor by the 1430s, and the Burgundians were starting to tire of their alliance with the English, which is particularly important since it was Burgundian troops who were garrisoning the majority of northern France.
I wonder if Philip the Good wouldn't end up quarrelling with Henry - possibly going so far as to demand a crown as King of Burgundy for his troubles, thus also effectively ending the question of whether he actually wanted to be Henry/Charles' vassal - and one way or another I see Henry losing his most important ally.
If Philip turns to the Dauphin's side, things could go very wrong very fast. If Philip simply pulls out of the war and decides to stay neutral while he waits to see who wins, I see Henry running out of troops and men and only being able to fight the Dauphin to a standstill. What happens next is something I would greatly like to explore, but don't feel qualified enough to answer.
The other problem Henry has is that he uniformly failed to get any of the French nobility on his side - and why should he have, I guess. While he filled some of these vacancies by stocking several of the larger estates with loyal English nobles, there frankly aren't enough English nobles to hand out every estate, and honestly it would probably be wisest not to create a situation where the majority of the English nobles have greater land holdings in France than England, anyway. He largely had the support of the burghers and the religious leaders, but this was partly power-play and largely fear. Without any noble French supporters he really is going to struggle to hold the land for large periods. To have a chance of creating a lasting Anglo-French Kingdom - even to the end of the century - he really needs at least a thick brush-stroke of complicit French nobles, both to fill every manor vacated by the fleeing French lordlings and to sooth the peasants when they start blaming the King for everything that goes wrong. Problem is, I'm not sure where he can get this support from. A century earlier, Edward III might have been able to get it because there was some genuine support for his claim - the King of France (can't be bothered to look up the name, sorry) actually had to exile or imprison the entire University of Paris, because the law students and teachers there debated the subject and decided that Edward had the better claim. By the reign of Henry V, the French nobles have been indoctrinated enough to the idea of French nationalism and anti-English sentiment that no-one really wanted to take the risk and swap sides.
