Loosen the small screw at the base of one of the doorknobs, using your screwdriver. Twist the knob until it comes off the treaded spindle.
Pull the other doorknob out from the latch, taking the spindle with it. Remove the knob from the spindle, by loosening the screw at the base of the knob and twisting it off.
Twist the new spindle into the base of one of the knobs, or of a new knob, if you are replacing them. Tighten the screw at the base of the knob.
Feed the free end of the spindle through the square hole in the latch assembly, until the knob is against the door.
Hold the knob steady while twisting the second knob onto the other end of the spindle, on the other side of the door. When the second knob is fully threaded onto the spindle, tighten the screw at the base of the knob. Both knobs should turn freely.