I think flowering plants with tall whippy stems are best avoided, esecially those with large daisy-like flowers, the sunflower would typify this. interestingly the word daisy has derived from days-eye, aluding to the fact that it opens up to the sun and follows it around.
If its a perennial border you have then I would seek out low-ish, shade resistant, ground-cover plants of woodland origin.
A few I think may do well are - Convallaria ( lily of the valley )'variegata' has beautiful cream and green leaves.
Bergenia (elephants ear) has large, leathery round green leaves with pink or white drumstick-like flower heads that are unlikely to be drawn to the sun.
Symphytum (garden comfry) will quite happily romp away all on its own, needing little attention.
I dont know if this is possible but is their any way you can improve the light from the shady side ie pruning ? but I guess you would have already thought of that. Good Luck.