Hello, Thank you for using this service, I will be helping you with your problem today
Good morning, if this goes through the wash and drains but just does not spin it means your motor coupling Has Broke.
this is a very common issue on these machines . .The belts were replaced in these with a direct drive coupling. .Most likely this is the rubber you found,these are not difficult and a very inexpensive to replace . you can order one at the link below and there is a video there also to show you how to replace it..
heres the link. http://www.repairclinic.com/PartDetail/Motor-Coupling/285753A/1195967?modelNumber=LSQ9564MQ0 thanks and best wishes, Bryan