How to get rid of static electricity on a plastic slide
As I understand it, I need to ground the slide to provide a way for the static electricity to reach earth. But since this is inside, that's easier said then done. 


Close. Your slide is plastic, i.e. made of a good isolator. Grounding one point of your slide won't help much; then, no charge can build up at that point, but that doesn't help anywhere else. 

So, you need to make the sliding surface conductive. Doesn't have to be a good conductor – just ever so slightly conducting that the small high-voltage charges will rather "flow away" to ground than make lightning.

Luckily, there's anti-static spray, especially made for e.g. casings of sensitive electronics. However, since your problem is a large plastic slide: I think there's cheap anti-static sprays for clothing. Clothing that needs that is either silk or synthetic, i.e. isomorphic to plastic slides ;) So try these.