Hi Martin, Yes, you have a Cordyline australis and now seeing the photo, I can see your problem ! I think i'd be inclined to cut it right down and start again from the base. Its a new one on me that you can root the tops as a cutting ????
If it were mine, I'd remove it completely and replace it with a dwarf fan palm (Chamaerops humilis) which would be more in keeping with your available space, its fully hardy (infact much hardier than Cordyline) Another option might be to grow a red Cordyline in a tub (which would have a slight dwarfing effect) and would not ultimately get as big.
regards Lannerman.