In this sentence, why is the 2nd "le" necessary?

Bonjour Peter !

That's an interesting case, as both le don't have the same meaning !
"Elle ne le fait pas, bien qu'elle le puisse."

The first one refers to the "thing" (le ménage for example) that she's not doing, whereas the second le refers to the action in the first part of the sentence, i.e. the fact that she could "do it".
Hence the need to have both.

See also the lesson https://french.kwiziq.com/revision/grammar/use-le-or-l-to-refer-to-previously-mentioned-complete-ideas

I hope that's helpful!
À bientôt !