I found the instructions in the link to be a little too confusing for me. I think it's easier if you understand that offset umbrellas have a single piece of rope that is tied in a ratcheting mechanism (turn handle) and pull at the tip of the umbrella to pull it open. When the string breaks the umbrella closes and is limp. The ratcheting mechanism allows you to hold it taut. The string connects at the tip through something that looks like a bead. It's only real function is to prevent the rope from slipping through. Therefore when you have a break you basically need to replace the whole string.
Most instructions tell you that if you broke the string you can use the piece of broken string attach it to a new string and pull it into the ratchet mechanism. The rationale for this is that it simplifies threading the string through the pole. It is only difficult to thread the string if there are a few turns (there's one usually at the end) and if there are any small holes to thread through. I found that there was one small hole at the tip, but none near the ratcheting mechanism and yes my new rope got stuck at the corner. The truth is that most of us don't know the rope broke and so we already pulled it all the way in. I found two things make it easier to put the new rope through.
1. You can get higher grade rope that is actually metal wire with plastic around it. Check at Home Depot and make sure you don't get something that is thicker than the current rope or at least not much thicker. The reason not to go too much thicker is that some of the holes you have to thread through may not accommodate a thicker rope. The higher grade rope/wire will be much more rigid and will thread relatively easily.
2. How do you get across the corner. Most of us think you have to start at one end and go to the other end. If instead you start at the corner and take one piece and thread it down and the other the other way you will find that you have turned the corner into a starting place and essentially eliminated it as an issue.
If you try to attach a similar or thinner rope to the existing rope you will be able to do the repair, but you won't be able to improve upon the existing rope. The reason it broke is that it's cheap and it wears quickly especially in the wind. Most good rope is rated about 40-50 pounds. That's not a lot and in the elements with water etc it will break again. If you learn to completely rethread without attaching to the existing rope you can find better thicker rope. I found at Home Depot they had a really nice wire rope with a plastic housing. It was rated 120Kg or about six times as strong as what was in there. The likelihood that this sucker will break is pretty low and I won't have to worry much about over tightening.
You will find that you can pop off various pieces to reveal the bead at the tip and in some places there are tips that remove that make threading easier. Always look for the seam it will make your life easier. Finally, take a picture of the ratchet mechanism right when you open it and as you pull it apart. I guarantee you that if you don't you will find all the pieces will come off quickly and fly all over the place. You will then have to go to the pictures from the links in the other emails to see if you can put it together right. It may take a few tries to get it right. Therefore its much easier to take the picture at each step so you can easily figure out the order of the bolts and nuts.