According to the LMS Help Desk, The token for Keep me logged in only lasts for four hours of idle time. The LMS recognizes idle time as no activity for 15 minutes. At this time, there is no way to make it last longer as it was designed more as a quick return item, e.g. if a Learner leaves and comes back in 30 minutes to finish what they were doing.
It would be helpful if the token lasts until the user reboots the computer or clears cache.