Bus/metro back toward the hotel (~15 min), then the trip's last-night laundry stop: El Safareig del Barri (self-service, Carrer d'Aragó 104 at Compte Borrell), a straight ~19-minute walk down Aragó from the hotel (1.4 km each way). Open 8:00-22:00, 365 days a year, +34 934 616 878. Wash and dry back to back, then the same walk back.
Call it two hours all in: ~15 min back to the hotel, ~19 min each way to the laundrette (38 min round trip), a 30-40 min wash and a 30-40 min dry running back to back. Nothing dirty gets packed for the 05:20 start tomorrow. Hotel valet is the backup if it's ever closed or you'd rather skip it, but self-service runs roughly €10-20 total for both of you versus $80-150+ for a two-person valet load.