This hadith is reported from Sayyidah Aishah (may Allah be pleased with her) through another chain, but it does not mention 'wahdahu la sharika lah' (He is alone, He has no partner).
And the narration from Sayyidah Aishah (may Allah be pleased with her) putting the words of salam first has already passed.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الصلاة / 2841