I think you need to do some more studying of your NT and of history and of maps.
There is no mention of an Inn nor of them seeking lodging in the NT. You and Ama are reciting the local Church Pageant version of the birth of Jesus. It is not in Your Book.
1:50 AM is generally referred to as "the early hours of the morning" or "very late that evening". It's not significant because much of Luke is mis-translated and I'm not familiar with the particular passage, anyway.
I think if you do some checking you'll find that "inn" is a mis-translation of "guest room". Joseph was heading back to his home town (if you believe the "go home to pay your taxes" variation) and they would have much more likely stayed with his relatives. There is repeated mention of manger, but no confirmation as to whether it was indoors or outdoors, in a stable, or the main part of the house. But other than that one mis-translation, there's no mention of an inn.
Again, do some checking. True Believers (NT Scholars, Professors of Divinity, History Professors, Authors) are aware that the distance is, as you roughly calculated, about 60 miles if you take the direct route. Unfortunately, though, the direct route would probably get you robbed or killed as it went the hostile territory of the Samaritans. The accepted route is that they would've trekked quite a bit eastwards and followed the river course. It would've been considerable longer than the helicopter route you found. (You did say they likely rode a donkey, right?)
Now, unlike the African Swallow, the known breeds of donkey at that time, could not do 24 mph. Joseph's humble donkey could do about 4 mph, unladen (maybe less - that's the calculation for big healthy modern donkeys). But with a big fat pregnant lady on board, maybe three mph, and more important than that,.... it DOES NOT MATTER the walking speed of the donkey. What matters, like in team time trials, is the last person to cross the line, and that would be Joseph - on foot. Human walking (strolling) speed is about 3 mph. A person out on a long trek is probably doing more like a long distance hiker, though, and that's more like 2.5 mph.
And you've got stops for breakfast, lunch, supper/dinner, stretch your legs breaks, pee pee breaks, etc... If, as you assume, Mary was birthin' that little Savior any minute, they might travel a maximum of five or six hours a day. The total, even at 3 mph, would be not more than 18 miles per day, or six days to get there (on the seventh day, the donkey rested - so sayeth the FSM).