Checked
these for closeness to your property, not much luck - I would ask at the hotel and businesses appealing to the public nearby (locally employed employees).
Near the Design Museum, however, is Laundry Box, 1 Shad Thames, London SE1 2PN; 020 7403 0808, 1.3 miles, south on Tower Bridge road, then east (your left as you care coming southward across the Thames). You are most likely looking for a "wash and fold" or "self service laundry" in the UK, iirc.