Hello, I'm Dr. Deb and will do my best to help with this concern about Clarke.
I'm so sorry that you and millions of others are being affected by Irma:(
If Clarke were my patient, I'd likely dispense Doxycycline if I suspected he had kennel cough; however, since you have Cephalexin, this is what I'd give him. The dose for a dog this size would be 500 mg twice a day, with food to avoid stomach upset.
As you probably know, kennel cough can be secondary to either a viral or bacterial infection so if this is a virus, it will have to run its course ...which is about 10-14 days.
I'd also feed him softened or moist food which may help to prevent further irritation to his throat. (Dry kibble can be soaked in water for 15-20 minutes until its the consistency of oatmeal.)
I'd probably discontinue the Temaril-P since the steroid component wouldn't be indicated for kennel cough.
I hope this helps and that you both stay safe, Deb