Repetitions help to conjure up the speaker's evasive voice.
The first stanza, for instance, ends where it began: the speaker starts by incredulously exclaiming "I tell my secret?" and ends by firmly declaring, "my secret's mine, and I won't tell." That roundabout refusal sets up a pattern for the whole poem: the speaker will keep on playing with her listener this way, reminding them that she has a secret and refusing to admit it, over and over.
There's a similar effect at the beginning of the next stanza, in lines 7-8:
Or, after all, perhaps there's none:
Suppose there is no secret after all,
The echo there hints that the speaker is enjoying her listener's attention, and she's trying her best to keep it fixed on herself. Having just refused to reveal her secret, she opens up a new corridor for exploration—the idea that she might just be a big liar—and then, gathering steam (perhaps as she observes that her listener's curiosity is piqued), repeats herself.
Some of her reasons for all this teasing might appear in a moment of polysyndeton in line 6:
Only, my secret's mine, and I won’t tell.
This is a tautology (a statement that restates itself, something obviously true). But it's also a really important feeling for the speaker. Her secret is her precious private possession—and it brings her power. Perhaps her delight in the idea that the secret is hers suggests that she doesn't feel she owns much else in the world; this secret is her one treasure.
Elsewhere, repetitions give the speaker's voice an impish, fanciful edge. Take the anaphora in these lines, for instance:
And let the draughts come whistling through my hall;
Come bounding and surrounding me,
Come buffeting, astounding me,
These bouncy repetitions suggest a speaker carried away by her own playfulness as she imagines something that isn't currently happening. Picturing cold winds frisking around her like badly trained dogs, she uses fittingly frisky rhythms.
In the poem's closing stanza, the speaker slows down a little. Here, repetitions suggest she's savoring her listener's torment, drawing their suspense out to painful lengths. "Perhaps" in the summer she might decide to reveal her secret, she says in line 28—but only "perhaps," she reiterates in line 33. Her anaphora suggests she relishes her secrecy so much she might play it out indefinitely. So does another moment of anaphora: her resolve to reveal her secret only if there's neither "too much sun nor too much cloud"—only if the light is just right.