Her: Ahhh I thought I recognised you 😉
Student: We’re getting a prenup and I’m keeping the dog though
Her: No you can have full custody of the kids that you begged me for, I’m keeping the dog and the waffle maker

You know the kids won’t leave the dog behind. This is why it never worked between us
 This message works as a continuation of the joke while also framing the student as the one who initiated the break up, putting him in the position of power.