You do have a 6 speed then as the shifters are not cross compatible, and it sounds like you have a selector issue.
1st thing I would try is to sit in the car without the engine running, depress the clutch, and pull up the reverse slider, then see if you can hit 6th gear. It's possible that there's been some wear and movement in the selector shaft, or the dowel pins half dropped out or something, and getting into 6th is restricted by the safety tab meant just for reverse.
If that doesn't work, I'd try being really brutal getting into 6th gear without the engine on, and if that doesn't work (which it almost certainly won't, but it's worth a go), it's time to remove the box and have a look inside.