I have the same washer, had the same problem. The washer will not spin fast and drain unless the lid is closed. When you close the lid, a small pin on the lid contacts a normally open toggle switch that allows the washer to spin and drain. This switch is a safety measure to ensure nothing and no one could stick their hand into the washer. The drum spins so fast that if, for example, the lid was open and someone stuck their hand in and their sleeve got caught on the spinning agitator, the washer could instantly twist, mangle, or severely injure the arm. In your case it is also possible that the metal tab protruding from the lid is deformed and not contacting the safety switch for some reason.
If you are daring or do not plan on stupidly opening the lid during the spin/rinse cycle then you may repair this problem AT YOUR OWN RISK:
The lid safety switch must be activated in some way, you can jam something in the hole so that the washer always thinks the lid is closed.
This is the repair I did personally for my Kenmore washer. I opened the unit to access the switch and wiring. There were multiple fractures, cracks, and missing pieces of the plastic switch housing in my unit so it was not salvageable. I discarded the broken housing and simply joined the wires together to create a permanently closed connection. I made sure to solder the connection, insulate the joint completely, and secure the wires to a safe mounting location away from any moving parts. My Washer still works today. If you are handy and do not want to pay a technician a few hundred dollars to replace the part you have this option.
BEST, MOST RELIABLE, AND SAFEST SOLUTION:
If you have the skills find the exact part number for the switch and replace it yourself or call Kenmore and schedule a repair service.