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if this spins freely on the transmission shaft even with a new set screw unfortunately the agitator is stripped and needs to be replaced. the agitator runs around $65.00 and I have included a link below where you can order it. this agitator doesn't have a replaceable drive block as some do.
click here for part , http://www.repairclinic.com/PartDetail/Agitator-Assembly/22001821/454575?modelNumber=LAT9416AAE thanks and best wishes, Bryan
the shafts on these don't strip because the agitator itself has plastic spines which cannot strip the transmission shaft. on the transmission shaft there is a grove the screw sits in to keep it from sliding up. you can see the grove in this picture, that is a normal grove.