I also have this problem... It randomly happens to me and I have tried to search for causes or things to help with the eye pain but I could never find anything. Let me know if you have better luck
Pete, you might want to keep a log so you can see if a pattern emerges as to if anything precedes these rough nights. Also, goggle sleep hygiene and see if there's some things you need to implement to provide the best opportunity for sleep.
Most people don't feel rested after 5 hours of sleep. If you are waking up after 5 hours, and it's not by an alarm clock, I'd be wondering why. You don't say how long this has been going on or how often, but if it's affecting your daytime life, maybe you could benefit from a sleep study to see exactly what's going on while you sleep.
Also, if you are on any meds, check with your pharmacist if this could be a side effect. And some people say a small dose of melatonin helps them fall asleep. Best wishes at resolving your sleep issues.