I recommend KIMS for washing, ironing or dry cleaning your clothes. The nearest branch is at Velyka Vasylkivs'ka street, 29 (map here), open 8 am - 9 pm on weekdays and 10 am - 9 pm on weekends. Normally, you can pick up your clothes after 2-3 days.
You can leave your luggage in storage lockers at Kyiv Railway Station or Boryspil Airport. Unfortunately, we do not have any special lockers in the apartment.