Depending on the soap you used, it may be worth doubling up. When I have run into this I have used all three of these in tandem:
~ White Ivory soap
~ Colorless hand soap
~ Regular blue Dawn Dish soap
The problem there is that these may dry out the hair more than expected, so using a nice natural (and odorless) conditioner may be necessary to return the sheen and detangle.
I stay away from baby powder, myself. Pure corn starch (straight from a local grocery store) is more natural and does not have a composition that should compromise the nylon hair due to light exposure or other chemical reaction. Use that and gently comb it through if you just want to reduce the tackiness, then wash after combed through. Only leave it in the hair if it doesn't leave a gritty deposit.
Nylon will also (should) withstand a spritz of 50% isopropyl rubbing alcohol. I would recommend spot-testing with the alcohol first because of the age of the nylon, but I would only recommend that as a last resort.