Cheapest new piano, a Chinese factory job, will set you back about £1800 minimum.
You could go down the electronic route, and could get a nice new upright for about £500, which will last your DD a good few years and won't need tuning.
Second-hand ones vary wildly in quality - if it hasn't been tuned then avoid, it may be untunable and warped. Go and see ones on EBay, Gumtree etc with a friend who knows about pianos before purchasing - a free piano may sound like a great plan but not if it's unusable and costs over £100 for someone to pick it up and skip it!