Nong Khai is a City in Thailand and shared border with Vientiane.
Muong Ngoi is in Luang Prabang province next to Nong Khai then you should spend at least 7days/6nights for duration. Hereunder is brief itierary:

Day 1: Nong Khai – Vientiane – Halfday tour
Day 2: Vientiane – FLT to Luang Prabang + Half day City Tour
Day 3: Luang Prabang – Nong Khiaw – Muang Ngoi by boat
Day 4: Muang Ngoi – Muang Khua ( boat)
Day 5: Muang Khua – Oudomxay – Luang Prabang by car
Day 6:Luang Prabang – Kuangsi Waterfall
Day 7: Luang Prabang – departure

Hope its will hepl!