Will this kit fit delta kitchen faucet model 978 SSSD-DST ?

Thank you for your interest in Danco products. Our Danco 86970, Single Handle 1-Handle Metal Faucet Repair Kit For Delta would not be compatible with the your Delta RP 50587 replacement cartridge.

This should be just what you need. To be one hundred percent sure, I would ask the Lowe's person in that dept. They are smart people. I am sure this will have what you need though. Hope this helps!