@ossobuco,
Thank you for all that lookagement!!!
It is the invisibility that confounds me so.
I might try some lemon essential oil on the balcony door...which I assume to be their means of invisible ingress.
I also plan to remove myself and the cat from each room in turn, then spray every crevice, then leave the room closed for a couple of hours.
As they are invisible I shall, of course, not know if my murderous intent has been effective...but I will feel a little better.