If you are going to get up through the night then we used the "no error" way of housetraining.
You have pup in a crate or box in the bedroom and everytime they wake you take them out and wait till they go. The first night or so its a total marathon but ours were clean in a week and sleeping through a short night at 10 weeks.
During the day take pup out on waking before food ,when she drinks, after or during play and every 30 minutes by default and STAY with her till she goes.
Reward with something totally yummy for going outside and ignore accidents, just quietly clean them up.
We just let ours run around outside, they hadn't been lead trained at that point and its an extra complication at 3 in the morning.
Clean the house thoroughly with something biological (either the purpose made stuff from a pet shop or a dilute biological detergent) Clean her usual spots especially carefully and clean again if she has an accident.
I know that people say get the pup to walk outside themselves and that's ideal if you think they can make it.
If you see pup start to squat or circle and sniff then I would counsell picking them up NICELY and carrying outside so they don't have a chance to make a mistake. I learned with this pair of pups from another website that if you need to hurry pup outside, then carrying them with tail curled under will cork them for that precious minute needed to get them outside and not get baptised!
Last point is that I too use a crate and pups will be very reluctant to soil it but WILL do so if left too long.
Now this will be very distressing for them and messy for you but also will teach them that yes you can soil in the crate which means that they are more likely to do it again...use a crate by all means, they are very useful, just don't push your luck!