@AxiomaticSystem posted the solution (without explanations) just when I was just finishing the step-by-step write-up, and we seem to have reached the same solution (phew!), so here you go:
The easiest place to start seems to be the 3x2 box. (I'll try to always mention width first, that should give unique names to all the golden boxes.) The 8 and 4 fix each other's positions, and we actually know the missing digit: there's a 2 in the bottom left corner, so we must fit five 2s into the 8x5 box. They can't all go in the 5x5 square part, because column 5 already has a 2, so there must be at least one 2 in the 3x5 box. And if there's one, there must be three, which again won't fit in the 3x3 square area. So the missing digit in the 3x2 box is a 2, and we get to start the puzzle.
There's only one digit missing on row 12, and again, we know what it is:
By the rules of the puzzle, the 13x8 box and the 8x13 box both must contain 8 sets of all the digits, so their contents are equal. Subtracting the 8x8 square from both, we get that the contents of the 8x5 box must be exactly equal to the 5x8 box. There's a 5 in the 5x8 box, so the missing digit must be a 5.
There are a couple of naked singles on row 11, so regular sudoku gets us most of the 8x5 box:
There may be easier break-ins to the other areas, but I managed to place the Q on row 7 first: it cannot be in the 5x3 box, because that would need two more Qs in that box, which is impossible, so column 4 is the only possibility. That leaves a naked single J in the same column. Applying the same login to the other Q and the 4s on row 6-8, we get a couple digits more. Also, the 3 and T in column 6 are forced into 2 squares exactly, which places the 6 in that column.
Now, the three digits that are in the 5x8 box but not in the 5x3 box must be 4, 7 and Q. (This information was available earlier, may I'll retcon my post later to pretend I had noticed it already.. :-) ). This means there must be a 2 in the 5x3 box. There's no room for two 2s in the 2x3 box though, so suddenly the 5x3 box is solved.
Now row 7 needs A9J, which we can place. The King in column 1 is naked (and single), and the 5 in row 5 is pushed into the 5x5 square, which resolves the row, and we're off and running.
As usual, the rest is pretty much just scanning the grid and hunting for singles, which is made not a bit easier by the presence of all the extra digits. I've included a couple of phases, mostly as a backup, in case I've made a blunder along the way.
And here's the final solution: