How long is a typical day in Dungeons & Dragons in terms of time in our real world? I'm using the 3.5 player's handbook, and I know a round lasts 6 seconds.
One day is 24 hours in-world, but an entirely arbitrary amount of time in the real world.
Player's Handbook v3.5, p. 146:
Complete bed rest restores 2 points per day (24 hours) for each affected ability score.
The words "day" and "hour" mean the same thing as they do in our normal world.
However, when playing D&D, of course, the DM can skip over multiple days with a single sentence, or a single 6-second combat round may take an hour of real-time to play, so this definition is only really important for in-world things like spell durations.
Time only passes in-game as the rules and DM narrate it to do so, and real-world time doesn't map to in-game time in any fixed fashion. For example, in-game time can essentially pause between game sessions. See this quote from the Dungeon Masters' Guide, p. 15:
It can be frustrating to DM and players alike that while in the game the characters continue what they were just doing, in real life the players have lived perhaps several days of real time between then and now.
In one D&D 3.5 game I played, a player had their character take a year of "downtime" to craft magic items. This took only a few minutes of real-world time. In that same campaign, we took a full hour to run one combat round which our characters would have experienced as an approximately six seconds, and once we paused a large battle at the end of a session so that while not even six seconds passed for our characters, a full week went by for our players.
Basing myself off a concept explained in "Nobilis", I think you made a small mistake in your approach.
When GMing, there are two types of "time":
"In character time": this is combat. Where time needs to be precisely calculated so you can know what happens when (ie: it's kinda tough to not track time if people use stuff like grenades, readied actions, etc. You need to know what happens first and in which order, so time is crucial)
"Out of character time": this is everything else, from traveling to sleeping. In these situations, you shouldn't focus too much on time passing, but rather on the actions everyone does. How boring would an RPG be if people needed to wait (travel time/6s per round) when travelling? You can just abstract that whole time.
Now if you want the math as to how many rounds in a day... I mean 86400 seconds in a day / 6 seconds per round = 14400 rounds. But you truly shouldn't be that pedantic about time outside of combat/time critical situations
D&D is a story. Passage of time in a story doesn’t correlate to time you spend reading it. Combat rounds often can take an hour IRL but are six seconds of time in the story.
A day in D&D can take multiple sessions (e.g. in a dungeon crawl), or go by in a second (“you sail for two weeks and reach Waterdeep”) depending on what’s going on in the game world.
Time moves as the narrative needs it. While in combat, a few seconds of game time can take hours to play. When your characters are talking with NPCs, time passes roughly in real-time. On the other hand, hours or days can pass in a moment if nothing interesting is happening.
At an extreme, we had a combat last several sessions (a bit of an excessive scenario), so in that case a day in game (actually, probably just a few minutes) took months in real time.