You have my sympathies. It is very difficult to kill morning glory and you need to be vigilant over a long period to completely eradicate it. As you have found, just spraying something like Roundup probably won't work.
You obviously need to get rid of all the roots but it is not easy as they can spread over a large area and pulling them out is not advisable as even if you leave a small part behind new vines will shoot. So cut back all the plants then use something like a blackberry killer on the roots and keep spraying any new shoots that pop up. I have also read that Weed and Feed for Lawns (MCPA and Dicamba) kills it but can't confirm.
If you can cover as much of the infected area with black plastic, weed matting or carpet it can also help smoother the plant.
Keep at it and I'm sure you will eventually get there.
Be careful with how you dispose of the cuttings, too. Don't leave them around, or put them in the compost. Best thing to do is to place into a plastic bag and dispose of in the rubbish.
Great to see you make your first post @Dave80. Many thanks for sharing that resource and for joining in the discussion.
Looking forward to reading about your own projects and plans soon.
Good luck with fighting it @Kezzlesj. As others have said, if you are persistent and keep at it I am sure you will eventually remove it from the garden and be able to transform it into something special.
Just thought I would also add a photo to this discussion so others can know what to look out for. I also note that the plant is still sold in some nurseries. I do not recommend anyone plant it!
Hope all is well.
Just wondering how the battle against your morning glory is going?