That can be really frustrating - but don't worry too much about it when it comes to your interview the next day. When I was interviewing, one time my flight was delayed on my first connection (I had 2 connections) and I didn't make my third flight. There would be no flights until the next day, so from my second connection location (which I arrived to around midnight) I drove down 9-10 hours. Arrived to the school around 10am, for an interview day that started at 9am (I got in around 930am, washed my face put on my suit - and with no sleep, straight to the school). I had notified them ahead of time of course.
Plus side? 1) Made for a good story to break the ice at times, 2) I tend to be super energized when I get no sleep (like some weird sugar high - until I crash later lol), and 3) they accepted me straight off the bat and that's the program I ended up going to. Later when I talked to the grad students, they mentioned that they had respected my motivation to get there no matter what and the fact that I didn't let it get to me.
So tonight, as someone else said, just get to your hotel - relax - and just remember that you're going to be okay. 🙂