So when lysosomes ‘recycle’ mitochondria they aren’t destroyed in that process as well, right? What happens to these hydrolytic enzymes (lysozymes) once they’ve done their job of recycling? Also, why would an organelle, such as mitochondria, be recycled by these enzymes?
Whether or not the lysosomes are destroyed is irrelevant really. In some instances their membranes will be completely voided (so yes, destroyed), others not so much. Mitochondria are recycled because they get, for want of a better way to explain it, get tired and a bit buggered essentially. Like anything, there's wear and tear on the mitochondria, so they need to be replaced.
The enzymes themselves may very well be broken down, or go onto recycling more things. Quite often they will be broken down though, or can be degraded or deactivated by other cell processes. They're all very nitpicky questions, particular about what happens with the enzymes and I daresay there's are questions we don't really yet have definitive answers to.