The book is giving you some screwy examples. "Let his game be played by him" is correct but weird. You would only say something like that to make very unusual emphasis.
Here are a couple things to know.
How to make the subordinate clause
To make the object of let into the subject of its own clause, you need to put it into the objective case and put the verb into the infinitive. That's why you say:
Let him play his game.
rather than:
Let he plays his game.
Similarly, you would say Let him be helped by us, not Let he is helped by us. However, this sentence is equally as weird as Let his game be played by him.
Two (or three) senses of let
I think what the book is trying to do is teach two different senses of let at the same time that it's teaching you a tricky form of passive voice. Two of the main senses of the word let mean: (1) allow/permit the clause to happen; (2) suggesting or agreeing that "we" do the clause.
A classic example of the allow/permit sense: Let me go! is what a person who is being held against their will says to their captor.
A classic example of suggesting that "we" do something: Let's go! or Let's get started! is what you say when you want to start doing what you and your listener were just talking about doing together. This sense nearly always has us contracted to 's. (It has to be us rather than we because the subject of the clause has to be in the objective case, as above.)
Those are the most common and simplest examples to remember in order to learn the sound of the language. However, those examples can't be converted to a passive form.
Here's a more-realistic example in both active and passive form:
Let Dr. Kildare see the patient.
Let the patient be seen by Dr. Kildare.
Possibly your book has confused the suggesting/agreeing sense with a third sense of let, expressing a wish. A classic example of using let to express a wish is: Let peace prevail on Earth. Here's a realistic version of what I think your book is trying to demonstrate:
Let us beat our swords into plowshares.
Let all our swords be beaten into plowshares.
or, passively again, without using let:
May all swords be beaten into plowshares.
I can see why your book might have confused these senses. They really are a big, muddy mess. They are all variations on the basic sense of allow/permit, stretched to mean different things by repeated usage. The wishing sense can often be understood as the allowing sense and the suggesting sense simultaneously, where the request/suggestion is addressed to a deity, like Oh, God, please let there be peace on Earth.
Is he the musician for the evening?
It's not possible to change the sentence into the passive as the verb be (is) is a linking verb.
You don't form passive structures with linking verbs like be, seem, become where the complement of the verb refers back to the subject. You need a transitive verb, which has an object, to form the passive. For example:
They requested me (object) to be the musician for the evening (active).
I was requested (by them) to be the musician for the evening (passive).
Best Answer
When you convert a sentence from active voice to passive voice, the subject becomes the agent (by...) or is omitted completely, and the object becomes the patient (the subject of the passive voice sentence).
The most common usages of passive voice are if
You use a reflexive pronoun when the subject and object are the same, so there is no point in focusing on the active-voice object, as it is the same as the active-voice subject. You do know who is the active-voice subject. The only remaining justification for using passive voice would therefore be if you did not want to say who was the active-voice subject. In this case, the second and third sentences become:
Note that the meaning of himself in the first sentence is somewhat different: in this case, the reflexive pronoun really means by himself or unaided or alone. To make this passive mood without specifying the agent, you would have to say something like:
Because by himself means alone, you cannot use himself as the agent without changing the meaning, for example: