Honeysuckle blooms on last year's wood, so don't prune it until this year's bloom cycle is done. Here's a link to tell you how and when:

http://www.ehow.com/how_5095740_prune-honeysuckle-bush.html

Here is a link that might be useful: How to prune Honeysuckle

Thank you for the response. I don't see anything that says I can prune back as far as I really want to...so I may end up just buying new plants and starting over. I really hate how it looks now, just bare vines up to the top. I guess I'll live with it this year, since I read to prune no later than mid-July. Maybe I should just find a vine that doesn't get that tall to fill in? Thanks again!

Your honeysuckle will be seeking the light. Since the lawn side is shady it will inevitably go up and search for the sun.

However, I can't see any advantage in buying and planting new plants when you could just cut down the current plants which are already well established. They will grow away again much faster than new plants. Then, as they start to throw out new shoots train them through the trellis to the sunny (patio) side and then horizontally back and forth across the patio side of the fence rather than letting them go up without check.

No honeysuckle will give you 'a nice thick screen of flowers.' You will get a screen of foliage with flowers in it. So even if you lose this season's flowers you will still have a green screen. Planting anew will not achieve this.

Oh thank you Flora, that will do nicely. So I can just cut them way back? That would be great. I'll wait until the late summer this year and see how it goes.

If you can kill honeysuckle, you are a smarter person the most of us! LOL This is one TOUGH plant!

Excellent! Just what I'd hoped to hear. Thanks!