With regards to how long it will take to fix this problem, it highly depends exactly what they're doing with the cable. If, as people have said, this is a full new cable that needs to be put in place then they will have to wait till the end of the season till any thing can be done.
You've got to remember that the cable for the Skyride is an absolute mammoth, they will have to carefully separate it from the Towers Street and Forbidden Valley stations then it will have to be carefully removed (or lowered? I'm don't know much with these things) without resulting in any damage to the areas and the Gardens. Then obviously it would be a nightmare of a task to get the new cable up safely (again, without damaging the Gardens/any areas) and then it has to be all connected to the stations systems and tested. Due to the sheer weight of the cable and the vast area it spans over it's not a simple matter of taking the wire down and getting a new one up one afternoon. It takes weeks to replace a cable and there's a lot of issues that have to be taken into account just incase they arise.
Realistically it would not be possible to replace the cable during the season without closing multiple areas of the park for a few weeks, although this would be a nightmare and would cause more issues than it's worth.
I'm sure if the Skyride went over a empty piece of land it would get fixed during the season, but with park areas, rides, members of the public and protected gardens all underneath the cable it's not so easy.