Catalyzes the phosphorylation of autoinducer-2 (AI-2) to phospho-AI-2, which subsequently inactivates the transcriptional regulator LsrR and leads to the transcription of the lsr operon. Phosphorylates the ring-open form of (S)-4,5-dihydroxypentane-2,3-dione (DPD), which is the precursor to all AI-2 signaling molecules, at the C5 position. (S)-4,5-dihydroxypentane-2,3-dione + ATP = (2S)-2-hydroxy-3,4-dioxopentyl phosphate + ADP + H(+) Belongs to the FGGY kinase family.