Hi, I have (had) a very large evergreen clematis that would get tons of white flowers every year. This winter was very hard on it. I started cutting off all the dead but there is so much I feel like I need to chop it down to the ground and see if it comes back.
The fact that there are still so many green leaves among the dead ones should give you hope that the vines are still alive. An easy way to tell is to scratch a thin layer of 'bark' off the vine and see if it is green just underneath. If it is, the vine is alive. By now I would expect buds would be developing too so look to see if you can find some. It's going to be a messy job to clean up all the dead leaves however and unless you choose to let them fall off on their own, you may want to prune the vine back somewhat along its length to make the job of tidying it a little easier. At least it's at fence height and not growing along the eaves. :-)
Too bad. You are obviously missing out on flowers and fragrance this year. It likely wouldn't be worth the trouble to extricate the living vines from the dead ones. As I'm sure you already know, Clematis armandii is a vigorous, fast-growing vine so, considering that it already has an established root, it should rebound quickly. Good luck with it!