A dynamic rope that sustains a fall factor that high should definitely be retired, even if it doesn't show any immediate signs of damage. A dynamic rope is designed to elongate (dynamic elongation of 30-40%) in order to adsorb the shock of such a high-force fall. By doing so it lowers the forces applied to your gear and your body (keeping gear from ripping out and body from serious injury).
Static ropes have essentially zero dynamic elongation ability. A factor 1.77 fall on a static rope would almost certainly rip gear, and cause severe bodily harm either in the catch or the corresponding fall. This is why you should never lead on a static rope.
As far as slings go, it's pretty cheap to replace the sling that caught the fall. You can easily re-sling cams and hexes. In my book it's worth the ease of mind for the next time I'm above that piece of gear.