The cycle can get stuck on the washer if the drain hose is not at the proper height. The end of the drain hose coming from the washer should be between 30 and 96 inches above the laundry room floor. Measure the height of the hose using a tape measure. If it does not fall within 30 and 96 inches above the floor, you need to move it.
How the drain hose is connected to the standpipe may be the cause of the stuck rinse cycle. The hose must have an air gap. If the hose is taped to the standpipe or fastened too securely, there is no air gap. It must be loose but securely in place to work properly. Using the hook that came with the washer to hold it in place should suffice.
If you have a high efficiency Kenmore washer and did not use HE detergent, you may have too many suds in the washer. When this happens, the washer will not advance to the spin or drain portion of the cycle. Instead, the washer automatically starts a special suds removal cycle to rid itself of the excess suds. It will not progress from the rinse cycle until this cycle is complete. Using HE detergent will prevent this issue in the future.
It is possible that the stuck cycle is due to the failure of an internal component. Try advancing the timer past the rinse cycle to see if the washer will continue. If so, you may have a faulty timer. Other internal components that can fail and cause this problem include the motor switch and wiring. A qualified appliance repair professional can access the internal components safely to diagnose the cause of the problem for you.