They are epithets. Specifically, they are stand-alone epithets termed antonomasia.
An epithet, according to the Princeton Encyclopedia of Poetry and Poetics, is:
A modifier specifying an essential characteristic and appearing with a noun or a proper name to form a phrase. (p. 452)
Greene, Roland, et al., eds. The Princeton Encyclopedia of Poetry and Poetics. 4th ed. Princeton: Princeton UP, 2012.
The key idea here is that an epithet denotes something essential to the person or noun it accompanies. The usual examples are grey-eyed Athena or wine-dark sea, both from Homer. Athena is inseparable from being grey-eyed; it is an immutable characteristic. Such characteristics need not be physical. Epithets such as Honest Abe for Abraham Lincoln or Mahatma Gandhi for Mohandas Gandhi furnish examples wherein non-physical characteristics are treated as essential to those individuals, to the point of being considered immutable.
When an epithet is used in place of aname it ordinarily modifies, it becomes an antonomasia. The Princeton Encyclopedia defines antonomasia as follows:
(Gr., “naming instead”). A figure in which an epithet or appellative or descriptive phrase is substituted for a proper name (e.g., “The Bard” for Shakespeare; “It was visitors’ day at the vinegar works / In Tenderloin Town” [W. H. Auden, For the Time Being]) or, conversely, in which a proper name is substituted for an individual, a class, or type (“the English Diana” for Queen Elizabeth; “Some mute, inglorious Milton here may rest” [Thomas Gray, “Elegy Written in a Country Churchyard”]). (p. 59)
Examples of the first sort of antonomasia, where a descriptive phrase stands in for a noun or proper name, include the Father of the Nation for George Washington, or the Light of the World for Jesus Christ.
When Sidney follows up his direct naming of Sleep with additional descriptors, any of those descriptors could be used by itself to mean "sleep". Sleep isn't a proper name, of course, but Sidney is personifying and apostrophizing it, which is what allows these phrases to be seen as antonomasiae. Shakespeare uses the same device in Macbeth:
Sleep that knits up the raveled sleave of care,
The death of each day's life, sore labor's bath,
Balm of hurt minds, great nature's second course,
Chief nourisher in life's feast. (II.ii.38–40)
Since phrases such as the balm of woe and Balm of hurt minds are complete noun phrases in themselves, they aren't ordinary epithets, which need both the adjectival and the noun they modify to be complete: Simple Simon or beautiful downtown Burbank.
It is true that the phrases Sidney uses are metaphorical: sleep isn't literally a knot of any kind. But antonomasia often are metaphorical. Washington did not literally beget the United States upon the mother country (another antonomasia). Christ is not literally a source of illumination. A camel is not literally a ship. While they are of course all metaphors, they are specifically antonomasiae.
The reason they are better described as antonomasiae than as metaphors is that antonomasia is the specific term for the use to which these metaphors are being put. Consider the term fair Ophelia. Here fair is an adjective, to be sure, but in context of its specific link to Ophelia, it is an epithet. So also with Sidney's addresses to Sleep: in the general case, yes, they are metaphors, but he is using them specifically as antonomasiae.
Being metaphorical is not, however, a requirement for antonomasiae. Chicago is literally a Windy City, because of the winter winds off Lake Michigan. Interestingly, there is an argument that the city is called that because its denizens are considered bloviators, and not because of the prevailing weather conditions; so perhaps that's not the best example. A better example is Maid of Orleans, unmistakably literal: Joan of Arc was a maid and from Orleans. It's worth noting that Joan of Arc is an epithet as well.
It is sometimes asserted that antonomasiae are formulaic, in the sense that their referents are generally understood. The City of Brotherly Love is widely recognized to be Philadelphia, or the Bard of Avon to be Shakespeare, to the point that the city or the poet themselves need not be explicitly named if these formulae are used instead. That would not be true of the prisoner's release or the baiting-place of wit, which aren't commonly-used phrases for sleep. If Sidney had not mentioned sleep, the argument goes, we could not say for sure what the prisoner's release signified; it could just as well be merely a key. So how can that not universally understood phrase be an antonomasia for sleep?
The answer is this: rhetoric is a matter of technique, not semantics. Any rhetorical tool is defined not by what it means, but by what it does. An epithet is defined as an adjectival phrase that specifies some constant characteristic of the noun or name it is attached to. An antonomasia is defined as a noun phrase that stands in as an epithet for another noun or name. Those definitions do not require that the epithet or antonomasia make sense regardless of context. On the contrary, like any other rhetorical device, those techniques make meaning only in context-delimited ways. For example, in the US, the Father of the Nation refers to George Washington. In India, the same phrase refers to Mohandas Gandhi. In Pakistan, to Muhammad Ali Jinnah. An epithet or antonomasia depends for its meaning on a context shared between the speaker or writer and the audience; it is not a contextless utterance. So in the context of Sidney's poem, he has established that the referent is sleep, and the remaining phrases are antonomasiac regardless of their non-formulaic nature.
Since Sidney does specify "sleep" just before the series of noun phrases that (in this context) mean sleep, it could be argued that the phrases Sidney uses are not antonomasiae, but part of an epithet. The 1st century CE Roman rhetorician Quintilian says:
There are some, however, who think that the epithet is not a trope, because it produces no change. Their reason is that an epithet, if it be separated from the word to which it belongs, must (if it be a trope) have some signification by itself, and form an antonomasia. Thus if we say, by itself, He who overthrew Numantia and Carthage, it is an antonomasia; if we add Scipio, an epithet; and an epithet, consequently, must always stand in conjunction with something else. (VIII.vi.43)
Quintilian. Institues. trisagionseraph.tripod.com/Texts/Institutes1.html. Accessed 20 Jan 2021.
Quintilian says that some rhetoricians do not believe that epithets are figures of speech, because they mention their referents directly rather than figuratively. "Joan of Arc*, for example, directly names Joan. Antonomasiae, however, are figurative because they allude indirectly to their referents: the Maid of Orleans stands in for Joan. If I say Joan, the maid of Orleans, however, I am directly naming Joan and so I'm no longer using antonomasia, but epithet. In this context, then, Sidney names sleep and then goes on to use various descriptors for sleep. So the entire first quatrain, barring the first three words, could be considered an epithet; or less strictly, since each phrase such as the poor man's wealth can stand on its own and isn't tied to the referent "sleep", the individual phrases can be considered antonomasiae.
Edit. Thanks to users Rand al'Thor and Chappo Hasn't Forgotten Monica for pressing me on certain claims so that I had to dig deeper to improve this answer.