I would go with the hardwood wedges as well, but I would also consider a screw and plug from the outside of the stringer. A bit of work, but it would be a long term fix. Only problem is that you can only get to one side. That said, I would pull them apart and start again. I don't think it would take more than a weekend.
The other option is to use a pocket hole jig underneath each tread and use two screws from each side. Again access wont be too easy on the lower treads but the pocket hole method is very strong, quick and easy. The jigs are cheap enough.