Short answer: At least $13.50 or $18 per $300 ticket, as a very rough guess. It varies depending on recoupment and that doesn't include agents' fees and any other royalties he may personally owe.
Longer answer: The way it usually works is the producer pays for the show entirely. This includes any clearances that may be necessary and any rent they may owe for the theater, plus cast and staff salaries. This can be anywhere from $5m to $20m+ for a 1-year Broadway run.
Then they start selling tickets. The the composer, lyricist, and bookwriter all share in the royalties with certain guaranteed minimums. They have a variety of Model Contracts with varying ways of calculating compensation.
I'm just going to figure based on what I understand about the Approved Production Contract for Musicals. In order to get these released you actually need approval from the Executive Director or Associate Director of the Dramatists Guild of America, but sometimes people will talk about them.
Not specifics, just generally. And usually in terms of the bare minimums. You can find the info on blogs and stuff sometimes.
So from what I understand pre-recoupment minimum royalties are 4.5% of ticket sales. This basically means until the producer has gotten his money back.
Post-recoupment minimums are 6% of ticket sales. Usually it's split three ways but in his case, he did all three jobs so he gets all three payments.
But Miranda may be making more.
Here's why, the much much longer version: A Broadway play goes in 4 separate stages. First it's what they call workshopped. This is like an off-off-Broadway version of the play with a bare minimum production. You'll sometimes see some big names get involved, because it's a way for them to work with the freshest of material, but they might get stuck spending 3 months doing some absolute travesty, and it's pretty unlikely that this cast will go much further anyway.
This falls under a separate contract, and it's possible that the theater that put it on is still getting royalties.
Then it goes into its Off-Broadway debut. This is a slightly higher production, usually a little closer to the final version but some of the major setpieces may be missing. This is also where a good number of the final cast and background players start to get involved.
This falls under yet another separate contract, and it's possible that the theater is also still getting royalties. But this may be offset if they're selling the soundtrack.
Then it goes to Broadway, where it enters Previews. For Previews, the audience generally understands that they're watching an unfinished production. Usually all of the elements are in place, but they're still ironing out all the kinks and polishing everything.
This is the beginning of the Broadway contract, but Preview ticket sales are counted different I think because you usually have a much larger backstage production team at this point.
Another main thing about previews is that critics aren't invited. They can attend, and they can even talk about attending, but they might piss someone off and not get invited to the opening if they say nasty things.
Finally the play is 100%, everything works flawlessly, all of the cast is locked in and you no longer need a 100 underlings to run around and put out fires.
So the play opens, and makes its run. Extended runs require a different contract, but it doesn't necessarily mean every year. A play that they are pretty sure is going to be a hit can start out as a contract for multiple years.
But you can't really go up to the writers and say "Hey, sign this three year deal. Oh, but it's minimum." They're just going to look at you sternly. But the producers have some arrows in their quiver, because most theaters are non-profits and ticket sales alone rarely cover the cost of production.
Top level writers rarely talk about their income in specific terms. They may generalize a bit but even that's considered low-brow behavior.
So at the end of all of that, the very very short answer is I doubt even the production's accountants have it broken down on a per ticket basis.