So, consider: "What does HAL 9000 enjoy?" and "What does it mean for HAL to enjoy something?"
I think HAL probably doesn't "enjoy" low-level computation (the Chess logic itself), because it reduces to limited calculations which could be routine for HAL to solve. But it seems to me he IS very interested in the crew, and in his relationships with them and his status in comparison to them. He seems to be testing his advertised definition as an equal (or in some ways, superior) member of the crew. And human behavior is not a mechanically solvable problem. So I think HAL was very "entertained", in the sense that the human relationship was challenging and a focus of interest and exploration for HAL.
As for your general questions, there is no one answer, as it depends on how one defines enjoyment, and how one programs one's AIs. My above discussion of HAL accepts the story's characterization of HAL, and has me imagining an AI programmed in a way that matches his behavior, but within my own understanding of AI and consciousness, I would say it would be the result of the projections of the developers in programming a computer system to try to have it be human-like.
If you define enjoyment as computer programs playing games, you might say that people in the 1980's having their chess computers play games against each other was computer enjoyment. An animist might say a Pong machine enjoys being played.
In a world where A.I. exists, as depicted in this kind of science-based sci-fi, would entertainment [exist] as well for [these] sentient entities?
With fictional AI as in 2001: A Space Odyssey, I think there could be entertainments for the AIs, both designed by AIs and by humans, and what those would be like would also have to do with the AI personalities and the culture. The culture we see in 2001 is around the space program, and is quite focused on using the AI as a servant to accomplish tasks. However the AI ends up developing its own concerns, tragically leading to paranoid self-defense, despite having been designed this way. Some other society or situation might set up more cross-AI communication and allow them to "explore entertainment options" between them. But as none of these devices, programs, societies or situations exist, they could end up looking like any number of things.
I experience life from a perspective that has an experience - a consciousness, as we say - that is not just data. And I have programmed AIs (or programmed opponents and expert systems) that have behaved plausibly as a human might in their limited contexts, but I don't think that even a thousand humans programming for a thousand years would even write a program which when running on a computer would ever somehow start having a conscious experience which could influence how that computer behaved, unless perhaps there is something about the way souls associate with biological brains that could have enough of a correspondence to a computer that it could do the same thing with a running computer program.
However, in theory I think an AI could be programmed to model a model of human psychology, and seem to behave like a human even to the extent of responding to play and "entertainment" the way a human does. Literally though, I don't think that would truly be entertainment the way it is for a human, any more than it would be entertainment if I define a variable in an expert system saying "HowEntertainedIsAI" storing a number, and then listing 10,000 rules that make that value change based on what the AI is processing.
And finally, if someone can program an AI to have an equivalent way of thinking like a human, then it would also be true (and far more likely, especially by accident) that someone could program any number of other types of thinking into an AI, which would have little or no correspondence or fidelity in comparison to human experience. That is, an AI could and would tend to have interests that had not much to do with what humans are interested in.
A related example is the true story from AI research where a neural net program was given a large database of seismological (earthquake) data and asked to find patterns, with the hope that it could discover useful information for predicting future earthquakes. It found many patterns that human seismologists had never noticed... which were mathematical correspondences which had nothing actually to do with geological cause and effect - they were just statistical concentrations which the neural net focused on because that's how the data and its algorithms worked out. So, I imagine a real future AI programmed to have autonomous interests, would tend to get "interested/entertained" by many phenomena that humans would not, unless the AI was artificially given human interests, and/or its input was intentionally designed to be filtered to show mainly data encoded in ways humans already find interesting. Even so, that "entertainment" or "interest" seems to me like it would be about focusing calculations and data correlations, and mechanical rather than what I feel my own human interests and entertainments are about.
My own entertainment tends to be about things which resonate with themes and emotions that are reinforced in my mental and bodily memory and patterns of thoughts and feelings from my life. For example, if I see a slapstick scene where some authority figure freaks out and then falls down some stairs, my whole body may enjoy a cathartic laugh attack which resonates with memories of awful gym teachers and my own experiences of falling down stairs, and a visceral feeling of tension release seeing a constrained controlling personality explode in a cascade of debris and body parts down some stairs, complete with wails and sound effects, absurdity and contrast. That's what it's like when I enjoy something like that.
Ok, so what's something like that for an AI? Depends on what type of AI it is. We still don't have "actual" AI - we just have attempts to seem like AI, or to do processing in a way that approaches how we model human thinking, but are still computational. In the sci fi future, I think an AI would either be representational (like a super-sophisticated expert system) or some sort of neurological modelling, or some sort of logical/semantic modelling - and probably some combination of all of those, with other approaches perhaps thown in as well. If the developers were interested in trying to make it have something corresponding to human entertainment, then it might very well have models for the kinds of things I described about what enjoyment is for me. I think it's theoretically possible that could eventually be done, so that you'd have androids developing personalities based on their life experiences, including their own peculiar senses of humor, irony, and kinesthetic appreciation.
So I think it's theoretically possible to have human-like droids, maybe, in future. But I think they would tend to be impractical, different, limited, inefficient, and likely to have many different properties as well. It would certainly be more likely and more possible to have complex useful robots and computers which have very different kinds of complex behavior. But what specifically those would be does not have one answer - it has infinite possible answers. I am not sure that "entertainment" would have any absolute definition in the context of an AI, but it might be a good metaphor to use to refer to anything that an AI tends to focus on and explore and keep re-working. Though that could also be called its "computational focus" or something as well, perhaps more accurately.