The taper is the same on both ball joints. The 5/8" ball joint will not normally fit up through the top end of the lower hole on a 9/16" knuckle. Terry has given you the best response on this--the bottom line sounds like you need the correct knuckles (spindles) if you want to stay with the bigger ball joints.
Even though I just posted this, I see your response, and I now have to ask what the ball joint studs looked like on the arms you removed from the car(?). If there is any damage indicated, oblonging of the hole(s) etc, the knuckle is toast--it won't take much eye-ballin' to see if a knuckle is damaged--look for displaced metal on the lower end, and look for the nut seat on top to be all chewed up.