How do I fix this wobbly crank and what do I need to fix it?

It looks like a bad bottom bracket given that the whole spindle is moving that much. The cranks need to be removed and bottom bracket needs to be replaced ideally with a sealed cartridge bottom bracket

You’ll need a crank puller tool, and the correct spanner/bottom bracket tool to remove the current one, and a new square taper bottom bracket. It’s most likely ISO English threaded. You’ll need to measure the spindle length to make sure you get the right spacing.