Mature kiwi vines are very much like a mature grape vine. They grow voraciously each year and can become a huge mass of woody vines if not pruned consistently. Don't lose heart, though, because they can be pruned back into a usable shape and will produce delicious fruit within one or two growing seasons. You can attack the mass of vines by following an organized process.
Remove all the winders. These are any branches that are wound around the trellis, other branches or just anything close by. This will be a huge portion of the mess and each cut should be at a 45 degree angle and about an inch out from the branch it is growing from.
Cut out all the cross branches. This might seem harsh pruning, but the vine will recover and be healthier in the long run. If you see any branches that are growing out and over other branches, cut them all the way back to an inch of the main stem. Realize that although you might be pruning severely on the top of the vine, the root system still has vigorous growth and will push out a flush of new growth in the spring.
Look for any suckers that are growing straight out. These will not bear fruit and should be removed. This will force the plant to push more energy into fruiting instead of supporting woody growth.
Pick the main stem for the vine and allow it to be about 6 to 7 feet long. Typically it is trained to grow straight up to a trellis and then a side shoot will grow over the trellis. Cut back any branches that are growing off the stem.
Allow two laterals to grow from the main stem after 6-feet of growth. These will grow across the top of your support. Cut them back to just three buds and remove any other laterals. Typically this is done in the winter after the vine has gone dormant.