Sprinklers can't be placed inside a greenhouse -- they can only be used outside. They cover a 3x3 area, but I think they can be upgraded to cover a larger area so you don't need as many.
I've decided I'm only going to keep the flowers and herbalism plants inside the greenhouse, while I do the fruits/vegetable/herbs (and trees) outside. Found a really nice build where I'm going to move my O'Dourke family; it had a "garage" but I'm not a fan of the fake cars so I converted that to a greenhouse with the glass roof and put in a few of the 2x2 planter boxes -- that should be plenty for my flower/herbalism plants.