The correct spelling of the phrase "didst the washing" is based on its phonetic transcription. "Didst" is pronounced as /dɪdst/, which is the archaic second-person singular past tense of "do". Meanwhile, "washing" is pronounced as /ˈwɑʃɪŋ/, which refers to the act of washing clothes or any other material. Altogether, "didst the washing" is correctly spelled using the phonetic sounds /dɪdst/ and /ˈwɑʃɪŋ/.