The speaker’s daughter was brushing the cat very lightly. Wanting to get more fur off since the cat was shedding, the speaker uttered this nice malaphor. It is a congruent conflation of “elbow grease” and “break a sweat”, both meaning to put forth a large effort or amount of energy to complete a task. “Work up a sweat” might also be in the mix as it also means to exert a lot of energy to complete a task. This one reminds me of one of my favorites: “Let’s roll up our elbows and get to work”. https://malaphors.com/2012/07/30/lets-roll-up-our-elbows-and-get-to-work/
A tip of the hat to John Kooser who realized he had uttered the malaphor, and then promptly sent it in.