Personally, if I was rebuilding a spindle I wouldn't be reusing the old bearings. I'd be buying new ones. Going thru that much trouble to get to the bearings just to reuse them again is in my opinion false economy. You must have taken the spindle apart because of noise or other problems. That noise and/or other problem(s) will most likely not be cured by a clean job and new grease. And when it comes to spindle bearings, cleanliness takes on the strictest sense of the word, which is difficult to meet.
Just my two cents worth.
Try to find yourself a syringe (I get them at the local surplus store) so you can accurately meter out the proper amount of grease into each bearing. The manufacturers literature will say how much is needed. The syringes small nozzle and the ability to meter are both in your favor.