Good on you for being realistic about the type of horse you are after, not many people seem to be able to do that!
Twelve years ago I sold my first horse (19 at the time) to a young rider for $1000. He had done pony club, been shown, jumped, was reliable on trails and very easy to float, shoe, clip etc and was the perfect horse for her at the time. Four years later she sold him to another person who just wanted a leisure horse and she got $1000 for him as well. Last I heard he is still in good health and being ridden occasionally, at age 31!
There should be plenty of horses between the age of 12 and 20 that wouldn't require frequent work and won't cost the world (i.e. $2000 and under). Just make sure they are in good health and it helps if you buy one that hasn't had too many different homes as you will be able to get a lot more information about their history.
Often the best horses for this type of thing are some of the mixed breeds like QH crosses, Arab crosses, welshie/coonemarra crosses etc.
All horses deserve, at least once in their lives, to be loved by a little girl.