What bike crank do I have and how can I re-attach?

Based on the new photos, there's a nut missing. Its been loose for a while, because the square taper hole is a bit wallowed-out. That may have killed the crank arm, have to try refitting it to be sure.

You need a suitable nut that fits the threads on your bike's bottom bracket axle, if the old one is not trapped in the head of the crank.

I'd pop off the silver cap in the middle, lightly grease the taper, then refit the crank arm. Then fit the nut and do it up tight with a socket - you can't use a spanner on this because its recessed.

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I suspect this crank has been loose for a while. You probably didn't cause this oritginally, but have exacerbated it by continuing to ride with the loose crank.

Tomorrow, retighten the nut again. The crank will slide harder onto the taper over time. Since its already damaged, you may as well try to save it.

If it breaks then you needed a new one anyway!

Check locally for a Bike Cooperative or similar - a left-hand crank arm, square taper shouldn't be uncommon. It may be that you need a new bottom bracket axle too, depending if there's damage to the taper.

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The reason yours has a (missing) nut and not a bolt is that its an older design, functionally the same. A more recent cartridge BB would have tapped holes in the ends for bolts, the effective differences are a tiny bit of weight saved, requires less space under the cap allowing for thinner cranks, and a smaller box volume.