Large Language Models (LLMs) allow social robots to engage in unconstrained open-domain dialogue, but often make mistakes when employed in real-world interactions, requiring adaptation of LLMs to specific conversational contexts. However, LLM adaptation techniques require a feedback signal, ideally for multiple alternative utterances. At the same time, human-robot dialogue data is scarce and research often relies on external annotators. A tool for automatic prediction of user enjoyment in human-robot dialogue is therefore needed. We investigate the possibility of predicting user enjoyment turn-by-turn using an LLM, giving it a proposed robot utterance within the dialogue context, but without access to user response. We compare this performance to the system's enjoyment ratings when user responses are available and to assessments by expert human annotators, in addition to self-reported user perceptions. We evaluate the proposed LLM predictor in a human-robot interaction (HRI) dataset with conversation transcripts of 25 older adults' 7-minute dialogues with a companion robot. Our results show that an LLM is capable of predicting user enjoyment, without loss of performance despite the lack of user response and even achieving performance similar to that of human expert annotators. Furthermore, results show that the system surpasses expert annotators in its correlation with the user's self-reported perceptions of the conversation. This work presents a tool to remove the reliance on external annotators for enjoyment evaluation and paves the way toward real-time adaptation in human-robot dialogue.