I remember "leker" because it sounds like "lake"...I imagine my younger brother and I as children, but we're giant....just to make it more memorable. We're throwing a ball across the lake, when the ball falls into the lake it creates huge waves that knock the boats around. We played like this so often as children, we created our own word for play: "leker"
"Å spille" is used for games with a defined set of rules (sports, video games, etc), while "å leke" is used for pretty much everything that doesn't fall into this category, and hence often what kids do.
Little kids "leker," and it can stand alone. You "spiller" a sport or musical instrument.
For instance 'she plays cards' would be 'hun spiller kort'. 'She plays hide and seek' would be 'hun leker gjemsel'.