If it works after pressing the button it's a mechanical fault. So you'll see when you open if the cog is sliding sideways on the shaft, turning independently of the shaft, or, if it has cracked again. I can't think what else it could be, except maybe a stray tooth of the previous cog jammed in the works - but that should lock up again on the very next rotation due to you 'opening a gap' for the piece to pass through by pressing the button... not work all day, then get stuck again the next day.
If it's not cracked but just sliding sideways, then the metal shaft is too 'slippery' and needs 'roughing up' (scratching with course sandpaper for example). And/or maybe any glue (if you used any before) has let go, but the cog is just tight enough that it works after you press the button.
Btw... Pressing that button while the car is moving was when many people said their speedo stops. Apparently it puts too much stress on a spring, against the cog, or something like that.
I've also read the grease Volvo used supposedly caused the cog to disintergrate. Maybe there's still some of that grease in there and it's finally caused the same result. Clean the area several times with cotton buds and metho (ok)/isopropyl alcohol (better) anyway.
If it's cracked another possibility is, I recall washing machine gearboxes had a common fault years ago (maybe still do)... where a nylon/white plastic gear attached to a metal shaft would shrink - but of course the metal shaft doesn't change size. So the nylon gear would split and no-turny. Maybe whoever is making the speedo cogs is using that same type of nylon/plastic and it's shrunk on the shaft and cracked.
