I'm sorry to hear your mirror frame got damaged. I agree with your assessment that it is made of lightweight foam. I suggest using Gorilla 59ml Glue Bottle as it is rated to join foam. I recommend using a light paintbrush to remove loose dusty foam residue on the join before applying the glue. Please make sure to find a safe place for the mirror while the glue cures. Our feline friends sometimes have too much energy. I suggest spending some brisk playtime with a laser pointer to taper off all that extra energy.
Please keep us updated, we look forward to seeing your mirror frame fixed.