You can take the IP to the guards, who can request what endpoint (router/mobile) was issued the IP at the time and date your Facebook was accessed. Note that the ISP may not release the info unless a court forces them to - this depends on the ISP.
Realistically speaking though, that's a lot of hassle and a pain in the hole for all involved.
From talking to guards, they tend to act proportionally to the damage caused. So they may not agree that your better half seeing you sending racy messages to someone else as a high priority, as an example.
Change your password and switch on 2 factor authentication. And don't walk away and leave your laptop/phone unlocked.
It's basic enough steps, but you're only trying to keep out people with basic enough knowledge most of the time.