If Kayle is fully empowered and has Svelsonger, does she gain extra Deflect and extra Might? I.e. +6 Might and +3 Deflect for each Svelsonger?

Yes — each Svellsongur gives Kayle a full second copy of her own text, so a fully Empowered Kayle (3×) gets +6 Might and Deflect 3 from herself, plus another +6 Might and Deflect 3 for each Svellsongur attached.

Her "+2 [S] per Empower" is +2 Might each (3 Empowers = +6), and Svellsongur's copied text is a separate active instance appended to her rules text (Rule 718.3). Deflect sums from multiple sources (Rule 809.2), so one Svellsongur = Deflect 6, two = Deflect 9. Might: 3 base + 6 (her own) + 6 per Svellsongur — 15 with one, 21 with two. Ganking stays redundant (Rule 810.2), and her "up to three times" Empower cap doesn't double — she's still capped at 3.