Answer: The quoted phrase ("Exhausted from the long, cold night") is a participial phrase.
Explanation: A participial phrase is introduced by a past participle or a present participle. Moreover, it can also include objects and modifiers. A participial phrase functions as an adjective because it describes and provides additional information about a particular noun or pronoun. "Exhausted from the long, cold night" is a participial phrase because it begins with a past participle ("exhausted") and it also contains modifiers. Furthermore, this participial phrase is working as an adjective: it is describing the noun "kittens".
